We need to process the problem: we have a CFG that originally could (perhaps) generate the target sequence, but terminal symbols were removed (some productions originally had terminals but they were removed, making generation impossible). We need to repair the CFG by inserting terminal symbols anywhere in existing production rules, with the objective to minimize the total number of terminal insertions required, to make the target sequence derivable from the start symbol A1.

We have 31 terminals (integers 1..31). The target sequence of terminals is:

1 1 30 27 13 29 24 26 25 1 12 31 28 27 23 17 15 16 19 20 18 1 14 21 22 17 9 7 10 11 5 3 4 1 6 8 2 7

Let's count length: There's 38 tokens? Let's count: I'll index.

1:1
2:1
3:30
4:27
5:13
6:29
7:24
8:26
9:25
10:1
11:12
12:31
13:28
14:27
15:23
16:17
17:15
18:16
19:19
20:20
21:18
22:1
23:14
24:21
25:22
26:17
27:9
28:7
29:10
30:11
31:5
32:3
33:4
34:1
35:6
36:8
37:2
38:7

Yes length 38.

Goal: add minimal term symbols to productions.

We need to think about the grammar structure. Many productions are duplicated heavily (e.g., A2 -> A61 A3 appears many times; A4 -> A5, A4 -> A18, A4 -> A19, etc.).

Our target sequence includes many terminal symbols that are present in some productions: e.g., rule 185: A5 -> 1, etc. So base grammar includes many terminals (1 repeated many times). The problem is likely that some productions that originally had specific terminals are now missing them. For example, rule 210: A9 -> 30 13 29 A25 28 27. This includes 30,13,29,28,27 ; that's exactly part of target sequence: after two 1's we have 30 27 13 29 24 26 25 etc. Let's see: rule 210: "A9 -> 30 13 29 A25 28 27". The target has "30 27 13 29 24 26 25". There's a mismatch: the production has 30,13,29 then A25 then 28,27. A25 expands later (maybe to terminal sequences). Let's examine A25: rule 278: A25 -> 1 ; rule 279: A25 -> A26 A25. Also A26 can produce 24, 25, 26 (rules 280-282). So A25 likely can generate a sequence of numbers 24, 25, 26 in some order? Actually A26 expansions produce terminals 24, 25, 26 - each as separate production. So A25 -> A26 A25 => can produce many combos of 24/25/26. Let's see: can produce sequences like 24 24 24 ... but also maybe different order when combine. Since A26 can be 24, 25, or 26. So A25 can generate any non-empty string over {24,25,26}. The target's segment "24 26 25" matches that: '24 26 25', which could be generated by A25 -> A26 A25 -> 24 (choose 24) then A25 -> A26 A25 -> 26 ... then A25->??? eventually produce 25 then end with 1? Wait A25 has rule 278: A25 -> 1 (i.e., terminal 1). So A25 can end with a terminal 1. But target after 24 26 25 there is "1". So sequence "24 26 25 1" fits A25: start A25, choose A26->24 combine with A25' -> then next A26->26 combine, then next A26->25 combine, then final A25 appears 1 (via rule 278). So that fits.

Thus A9 -> 30 (13) (29) A25 (28) (27). Actually A9 -> 30 13 29 A25 28 27. So after generating A25 which yields (maybe 24 26 25 1?) then we get terminal 28 then terminal 27. This yields "30 13 29 24 26 25 1 28 27". But target expects "30 27 13 29 24 26 25 1 12 31 28 27 ..." Wait the order doesn't match. Let's carefully examine target after first two 1's: target[3..] = 30 27 13 29 24 26 25 1 12 31 28 27 ... So after 30 there is a 27 before 13. Production in A9 has 30 13 29 ... differing order.

But there might be other productions that yield the order: maybe A9 could also be recursively used after some others that produce 27 at appropriate position. Or maybe there is a missing terminal inserted somewhere to reorder them.

In rule 210, the 27 appears after 28. Actually: ... A25 28 27. That's at end.

Thus the pattern "30 27 13 29 ..." cannot be derived as is.

Consider other productions: rule 450: A32 -> 30 27 A25. This looks promising: "30 27 A25". That's exactly the sequence we need: after "30 27" we want A25 (which yields 13 29 24 26 25 1 maybe). But we also need 13 and 29 etc. Let's verify: rule 450: A32 -> 30 27 A25. Indeed, that yields 30, 27, then whatever A25. The target after "30 27" is "13 29 24 26 25 1 12 31 28 27". So perhaps A25 can produce "13 29 24 26 25 1 12 31"? But A25 currently only generates numbers 24,25,26 and a terminal 1 at the end. So missing 13,29,12,31. It might have to produce them via further expansions (like combine with other nonterminals that generate those numbers). But A25 currently only expands to A26 A25 or 1. A26 generates 24/25/26 only, not 13 etc. So can't produce 13 or 29,12,31.

But there is rule 210: A9 -> 30 13 29 A25 28 27. That's a different ordering but includes 13 and 29 and A25. So maybe the derivation is structured as: we start with some nonterminal that expands to produce 13 29 after A5 or something else.

Actually note also rule 286-... many A27 -> A28 (so A27 expands to A28 uniform). Then rule 319-... many A27 -> A41. So A27 can generate A28 or A41, etc. There is also rule 319: A27 -> A41 . There is also rule 355: A27 -> A42 etc. Many alternatives. So A27 can generate many possible sequences.

So the grammar is quite flexible.

But the target sequence includes many numbers: basically a list that could correspond to a specific path through the grammar. We need to insert missing terminals into productions to make the target derivable.

Observation: many productions contain terminal numbers already: e.g., rule 210 includes 30 13 29 28 27 ; rule 450 includes 30 27 ; rule 474: A37 -> 23 A23 14 21 22 ; rule 475: A37 -> 23 A23 17 ; rule 474 includes numbers 23,14,21,22. There is also rule 474 that yields these numbers with A23 in between. A23 is expansions that produce 1 with further expansions maybe containing other terminals? A23 expands to 1 or A24 A23. A24 expands to terminals 18,19,20. So A23 can yield sequences like (18/19/20)* then maybe 1. So we can generate 23, 18,1 etc.

But the target includes "23 17 15 16 19 20 18". That order: 23 is present, then 17,15,16,19,20,18. Might be generated via other productions.

Edge: In rule 474 we have "23 A23 14 21 22". That's different.

We see many numbers in productions that might appear precisely once: 12 appears only in rule 780? Actually rule 780: A66 -> 12. So A66 can generate 12. Good.

31 appears maybe nowhere else? We see terminals up to 31: Many numbers like 31 appear only in rule "A66 -> 30"? Actually rule 784: A66 -> 30. That's 30. 31 appears as terminal perhaps from a production: maybe rule 742? No those are 1. 31 as terminal appears possibly only in rule 71? Let's search: May be some productions like "A66 -> 30" but 31 appears as number maybe in some productions later like A9-> ... Actually rule 210 includes terminal 30 but no 31. There's rule 455: A33 -> A34 A32 (has no terminal). rule 459 etc.

Actually maybe A31 produce 31? Let's check: A31 -> 1 only. Not 31. So perhaps 31 appears nowhere in the current CFG, which would require insertion. Indeed, target includes 31. The only production containing 31: I don't see any "31" within the list of productions given except maybe other parts: In rule 45? No. Let's quickly scan for "31". There is 31 in A56? No. We have rule 211? Wait rule numbers up to 31 for nonterminals. But in the list, there is "30 27" etc. I suspect the only numbers beyond 30 encountered are 30,31 etc. Let's check quickly all productions for terminal numbers:

- rule_210: 30 13 29 A25 28 27
- rule_450: 30 27 A25
- rule_474: 23 A23 14 21 22
- rule_475: 23 A23 17
- rule_472: maybe includes some numbers? I need to scan: Among rules for A36, A35 etc.

- rule_485? no numbers.

- rule_466: A35 -> A35 A35 (no terminals)

- rule_468: A36 -> 1 (no numbers beyond 1)

- rule_469: A36 -> A28 A38 (no numbers)

- And also many also have "23 17 15 16 A23 17" rule 702 (A54) includes numbers 23,17,15,16,... etc.

- rule_474 includes 23,14,21,22.

- rule_475 includes 23,17.

- rule_480: A38 -> 1 (no numbers)

- rule_484: A38 -> A41 A38 (no numbers)

- rule_487: A39 -> A39 A29 (no numbers) etc.

- rule_506: A43 -> 9 10 11 A21 7

- rule_730 maybe includes numbers: A59 -> ... not.

- rule_733: A60 -> 9 7 10 A21 8 7

- rule_836? Actually no.

There are also rules: A61 ->... doesn't contain terminals (except 1 from A62). So 31 appears only via A66 maybe? Let's see A66 rules: A66 -> 1 (multiple), A66 -> 12, A66 -> 30. So no 31.

What about nonterminal A66? Not generate 31.

The target includes terminal 31 after 12. 12 is from A66 -> 12. Then target token 31 appears. 31 is maybe generated by any rule like "A66 -> 30"? Not. But there is A1.. Actually there is "24 31"? Not.

Let's search thoroughly: In the whole production list, includes numbers:

- rule_210 includes 30 13 29 ... 28 27
- rule_450 includes 30 27
- rule_474: 23 ... 14 21 22
- rule_475: 23 ... 17
- rule_702: 23 17 15 16 A23 17
- rule_474,475,702 all include "23" and others.
- rule_738 (A61) maybe no.
- rule_740... no.

- rule_506: 9 10 11 ... 7

- rule_733: 9 7 10 ... 8 7

- rule_477: A37 -> A31 A37 (no numbers)

- rule_474, r541? Actually R???.

- Also rule 748? Not.

- rule 468: 1 etc.

- rule 629? Not.

- rule 562 included? No.

- There are also rules: A45 -> ... etc. No.

Search for 31 precisely: I see "A6 -> A5 A16" etc. A31 maybe includes "A30 A32". No.

Maybe 31 appears as part of some nonterminal expansions like A66 -> 30, not 31.

Potentially 31 is missing; we may need to insert it into some rule.

Similarly, 12 appears in A66 -> 12. So 12 is already producible.

Terminal 2 appears somewhere: A71 -> 2 (both 2 and 7). Good.

Terminal 3 appears? A22 -> 3. Good.

Terminal 4 appears? A22 -> 4. Also A22 -> 5.

Thus numbers 5 appear maybe via A22 ->5.

Terminal 6 appears? Not sure: A? maybe via A61? Actually A71 -> 2 or 7. A70 maybe produce 6? No.

Terminal 6 appears as part of A61? Let's search; I see "A72 -> 6". Yes rule 831: A72 -> 6.

Terminal 7 appears: A71 -> 7; also A71 -> 7 etc. So 7 is present.

Terminal 8 appears? Possibly not: let's search: I see "A59 -> ???" maybe not. A61 maybe? No. There is rule 505: A43 -> 9 10 11 A21 7, not 8. A60 -> 9 7 10 A21 8 7 includes 8.

So 8 is present.

Terminal 9 appears many times: e.g., in A43, A60.

Terminal 10 appears: same. 11 appears: same. Terminal 12 appears.

Terminal 13 appears: in rule 210.

Terminal 14 appears: rule 474 includes 14.

Terminal 15 appears: rule 702 includes 15.

Terminal 16 appears: rule 702 includes 16.

Terminal 17 appears: many times (multiple productions), e.g., A67 etc., or A71?

Terminal 18 appears: rule 274? Actually A24 includes 18. So yes.

Terminal 19 appears: rule 275: A24 -> 19; also A23 -> A24 A23 etc.

Terminal 20 appears: rule 276: A24 -> 20.

Terminal 21 appears: rule 474 includes 21.

22 appears: rule 474 includes 22.

23 appears: many places (A37, A54, etc.)

24 appears: rule 280: A26 -> 24. Also A56? Not.

25 appears: rule 281: A26 -> 25.

26 appears: rule 282: A26 -> 26.

27 appears: often via many rules (like A41 -> ...). Also A27 etc.

28 appears: rule 429: A28 -> A29 A28? hmm. But also many where A... Actually A45? Not. But they produce 28 in various combos may be missing. Wait in many rules, 28 appears as terminal? I don't see a rule that directly has terminal 28; it's in rule 210 as a terminal after A25. Also rule 212 maybe? No. Also A45 maybe? Not.

But 28 appears as a terminal in rule 210. Also maybe in rule 431: A28 -> A36 A43 (no terminal). 28 maybe not many.

29 appears in rule 210 as terminal.

30 appears many times: rule 450, 210, A66 -> 30.

31 is missing.

Thus the only missing terminal is perhaps 31; maybe also 6? But 6 present via A72. Let's verify generating 6 can be produced: A71 -> maybe 6 not; A71 only 2 and 7. A72->6 (as per rule 831). So 6 is present.

Thus likely we need to insert only "31" somewhere.

However also maybe 8 appears only in rule 733 (9 7 10 A21 8 7) and also possibly anywhere else. So okay.

Thus the minimal number of insertions may be 1: add terminal 31 to some production that appears in the derivation to match the target after 12.

Let's test if we can derive target with current grammar except for missing 31.

Given that the target sequence appears to be a concatenation of several base phrases that the grammar can produce (with variable expansions). Let's try to derive systematically.

Goal: Starting from A1, we need to ultimately produce the target sequence in terminals. Let's attempt to parse the grammar thoroughly. Our only start rule: rule_1: A1 -> A2. So start from A2.

There are many duplicates of rule "A2 -> A61 A3" repeated many times (rule_2 to rule_38): repeated 37 times? Actually there are many identical rules for A2 -> A61 A3, from rule_2 to rule_38 maybe. Let's count: there's rule_2 through rule_38 (maybe up to rule_38?), but note rule_38 is A3 -> A4, not A2. Actually rule_2 to rule_38 includes many repeated versions of A2 -> A61 A3. The list shows rules 2-38 maybe: Let's check list: rule_2: A2 -> A61 A3
rule_3: same
...
rule_38: A3 -> A4. So the last "A2 -> A61 A3" is rule_37? Actually rule_37 is also A2 -> A61 A3; rule_38 is A3 -> A4. So there are many identical A2 productions; essentially A2 always expands to A61 A3, but maybe they purposely repeated many times to allow for multiple usages, but they are all the same.

Thus we have A2 -> A61 A3.

So start: A1 -> A2 -> A61 A3.

Now we need to derive the target from A61 followed by A3.

A3 has productions:
- rule_38: A3 -> A4
- rule_39: A3 -> A27
- rule_40: A3 -> A44

Thus we can choose A3 -> A4, A27, or A44.

Now we need to produce the whole target from A61 A3.

Let's now analyze A61.

Production rules for A61: rule_738: A61 -> A63 A62
rule_739: A61 -> A66 A64
rule_740: A61 -> A69 A67
rule_741: A61 -> A72 A70

Thus A61 can expand to four alternatives. The terminals in these expansions: A63, A66, A69, A72 etc. So we need to produce start of target "1 1" possibly from A61 expansions.

Examine each:

A63 expansions: rule_758-760: A63 -> 1 (three times). So A63 can produce terminal 1 only.

A62 expansions: rule_742-757: A62 -> 1 (multiple). So A62 = 1.

Thus via A61 -> A63 A62, we get "1 1". That matches the start of target: "1 1". So likely we need to use A61->A63 A62.

Thus A61 expands to "1 1". Good.

Now after A61's "1 1" we have A3 remaining. Since the target continues after the first two 1s with "30 ...". So we need A3 to derive the rest: "30 27 13 29 24 26 25 1 12 31 28 27 23 17 15 16 19 20 18 1 14 21 22 17 9 7 10 11 5 3 4 1 6 8 2 7".

Thus A3 must generate that whole subsequence.

Now we have possibilities: A3 -> A4, A27, or A44.

We need to generate a long sequence that includes many numbers. Let's examine A4.

A4 has many productions expanding to A5, A18, A19, A20 many times repeated. There are many duplicate rules for A4 -> A5, repeated many times (like 61 to 70). Also many for A4 -> A18 (rules 77 to 112). Then many for A4 -> A19 (113 to 148). Then many for A4 -> A20 (149 to 184). So A4 can expand to A5, A18, A19, or A20. Since these have productions that lead to different terminal sequences.

Thus perhaps we need to use a sequence of these expansions to produce the target.

Alternatively, A27 may also generate some of the string; but we'll see.

Now examine A5: rule_185: A5 -> 1. Also additional rules: A5 -> A5 A18 (186), A5 -> A6 A5 (187), A5 -> A9 A19 (188), A5 -> A13 A20 (189). So A5 can generate various things, importantly includes the production A5 -> 1 which yields a solitary 1. So not used for complex sequences.

A18: has rule_252: A18 -> 1. Additional rules produce more complex stuff. It could possibly generate the '12', '31', etc.

Similarly, A19: rule_257: A19 -> 1. Additional rules produce complex.

A20: rule_262: A20 -> 1. Additional rules produce many complex sequences (including 9 7 10 A21 6 7 etc). So perhaps much of the rest of the target will be generated by A20 expansions.

Thus A4 can become A5, A18, A19, or A20. We likely need a combination: maybe A4 -> A20 is the big chunk that contains many numbers.

But the target after the first two 1's begins with 30, which is not a terminal directly but appears in productions such as:

- A9 -> 30 13 29 A25 28 27 (with 30 prefix)
- A32 -> 30 27 A25 (prefix 30 27)
- A66 -> 30 (under A61's other options)

Also A66 -> 30 appears as a direct terminal 30 under A61 -> A66 A64 alternative.

But we already used A61 -> A63 A62 to produce 1 1. We cannot produce 30 further from that, but later after A3? Actually we need to produce 30 after the two 1's; that could come from A4 -> A20 perhaps producing a phrase starting with 30.

Let's examine A20's productions:

rule_263: A20 -> 9 7 10 A21 6 7
rule_264: A20 -> A10 A19
rule_265: A20 -> A11 A5
rule_266: A20 -> A12 A20
rule_267: A20 -> A20 A18

Thus A20 can generate a lot, but not 30 directly, except maybe recursively through other nonterminals that produce 30.

Look at A10: productions include 1,A10 A8, A11 A9, A12 A10, A20 A17. So A10 can eventually produce A9 or others.

A11: similar.

Thus to find 30 in target, presumably we can get it via A9 or A32 or A66 or maybe also via A50 (?), but not.

A9's production includes 30,13,29,... that matches the beginning of the target after some reordering.

Let's find a path that yields sequence "30 27 13 29 ..." while also getting "24 26 25 1" via A25 somewhere.

The target segment: after the two 1's, the sequence is:

30 27 13 29 24 26 25 1 12 31 28 27 ... Then further.

Wait, actually after two initial 1's, we have "30 27 13 29 24 26 25 1 12 31 28 27 ...". The segment after "30 27" we see 13 29 then some of 24 26 25 1 (these come from A25). Then 12 31 then "28 27". So we need a production that yields "30 27 13 29" then needed A25 (24 26 25 1), then something that yields "12 31", then 28 27.

We have A66 -> 12 or 30. So maybe A12 (nonterminal) yields 12? Wait A12 is a nonterminal, not terminal 12. But A66 -> 12 (terminal). So perhaps we need to use A66 to generate 12. And 31 is currently missing; maybe we need to insert "31" somewhere near A66 production.

Let's examine A20 -> A10 A19 (264). A10 may produce "30 27 ..." via expansions that use A9 or something.

We need to see if there is a rule that yields "30 27 ..." that also can combine with A66->12 etc.

We need to explore the possible derivations for A4 -> A5, A18, A19, A20 chain.

Because A4 can be repeated many times (like 70 replicates). So we could have A4 produce a string of multiple nonterminals in sequence? Actually each A4 production yields a single nonterminal (A5 or A18 etc). So to produce a longer sequence, we need to recursively expand them using other rules like A5-> A9 A19 (gives two nonterminals in sequence). So combinations.

Let's try to design a derivation path piecewise:

Goal: target: 30 27 13 29 24 26 25 1 12 31 28 27 ...

We need to get "30 27" from perhaps A32 -> 30 27 A25, but then you would have "A25" after that, which is the sequence we want: "13 29 24 26 25 1"? No, A25 only yields 24/25/26 and 1, but not 13 or 29.

Alternatively, A9 -> 30 13 29 A25 28 27 yields "30 13 29 ... 28 27". That includes almost everything except we need 27 early after 30, but we have 13 after 30 not 27. So the target "30 27 13 29 ..." doesn't match.

But maybe we can get "27" from preceding symbols before 30? But we already have "1 1" and then we could have A4 -> A20 -> A10 A19 -> etc. Perhaps the 27 could be generated by something else (like A27 nonterminal). Actually there is a nonterminal A27 that can generate many things, including perhaps terminal 27 via expansions. For instance, many productions A27 -> A28 (which may eventually produce terminals?), but the base terminal of A27 maybe is built via expansions that contain terminals like 27? Not directly. However, there may be productions for A41 etc that generate 27 as terminal? Actually I do not recall any production that yields terminal 27 directly. Let's see: Rule 452: A32 -> A36 A33; later expansions may yield 27 as terminal via A41? A41's expansions include A38 A43, A39 A28, A40 A42, and A41 A41 (no terminal). But maybe A43 includes terminals 9,10,11, etc. There is no direct terminal 27 except as part of terminal sequences of longer productions (like in rule 210 includes "27").

Thus the terminal "27" appears only in productions as terminal literal in some sequences. For example, in rule 210, 27 appears at end. In rule 450, 27 at end after 30 27; appears after 30 27. In rule 474, there is no 27. In rule 475, etc.

Where else? Many rules include 27 as terminal: perhaps rule 459? Not. So the only productions that include the literal 27 are rule 210 and rule 450 (and maybe also others like A31 -> A7? Actually no). Also maybe A61 expansions have 27? Not.

Thus we need to achieve a literal 27 in target; likely we will use either rule 210 (which yields 27 at end) or rule 450 (which yields 27 at end). Additionally, there might be rule 474... includes 27. Actually rule 474 includes 23 A23 14 21 22, not 27.

Also rule 534? No.

Thus to produce terminal 27, we must eventually make use of a production that includes 27. Two known productions produce 27 at the end after other symbols: "A9 -> 30 13 29 A25 28 27" and "A32 -> 30 27 A25". In the second, 27 is second term, not at end. Wait rule 450: A32 -> 30 27 A25. This yields 30,27,A25 (which then produces other zeros). So 27 appears early, not at end. That might match target: 30 27 ... where after 27 we need more stuff from A25 (maybe 13 and 29 etc). However, this rule's A25 yields only 24/25/26 and 1. So not 13 and 29.

Thus we might need a more complex chain: maybe A32 leads to something else that yields 13 and 29. Let's see expansions of A25 we covered.

But there may be other ways to produce "13 29". Let's search: A13 is a nonterminal that may generate 13 as terminal? Actually rule 226: A13 -> 1, not 13. No direct terminal 13. However, terminal 13 appears only as part of productions like rule 210's sequence and rule part of different? Also maybe A64 includes 13? Not. So the only source for terminal 13 appears to be rule 210 (or perhaps other productions where 13 appears as a literal). Let's search throughout the grammar for "13". We have:

- rule_210: A9 -> 30 13 29 A25 28 27

- Possibly other rules: maybe A9 expansions other than rule 210 include "A9 -> A5 A17" etc., no constants.

Do any other productions contain 13 as terminal? Let's scan the provided list quickly: I saw rule 471: A36 -> ... includes "A36 -> A28 A38" etc., no numbers. rule 210 we saw; rule 214: A10 -> A12 A10 has no numbers. Later in the list: rule 216: A11 -> 1. Not. rule 272: A22 -> 4 (approx). rule 285? Not. Actually there may be other productions with constant numbers: e.g., rule 474: 23 etc; 475: only 23,17; 481: ???; etc. I haven't seen "13". Let's search systematically maybe. The guess is 13 only appears in rule 210; also maybe in rule 283? "A27 -> A41"? No. So yes.

Thus terminal 13 appears only in rule 210. Similarly terminal 29 appears only in rule 210 (and maybe 24,26,25 appear in A26). Terminal 30 appears in rule 210, 450, A66.

Thus to include 13 and 29, we need to use rule 210: A9 -> 30 13 29 A25 28 27.

Now we notice that rule 210 yields "30 13 29 A25 28 27". Not exactly the target order (need 30 27 13 29 ...). But perhaps we can reorder via inserting a terminal 27 before 13? Not allowed to reorder symbols; we can only insert terminal symbols anywhere. So we could "insert" a 27 before 13. But we cannot delete the existing 27 at the end. However, we can add terminals anywhere: before 13, after 13, before 29, after any terminal or nonterminal. So we could turn the rule into "30 27 13 29 A25 28 27". By inserting "27" after 30 and before 13. That would match target's need for "30 27 13 29". The original rule already has a trailing 27 at end, which matches final 27 after 28. So this insertion would give the correct ordering.

Thus we can modify rule_210: A9 -> 30 27 13 29 A25 28 27 (by inserting a terminal 27 after 30). That would add one terminal. This matches target's part up to that point. Let's see: after this production, we need after A25 to produce "24 26 25 1". A25 can produce that via expansions. Then we have "28 27" after that, which matches target's "... 28 27". Then after that we need "23 17 15 16 19 20 18 1 14 21 22 17..." etc.

Thus after that, we may need to produce the rest of the sequence via other nonterminals.

Now after A9 produces that piece, we need to perhaps produce "23 17 15 16 19 20 18 1 14 21 22 17". Let's see which productions may produce a "23 ...". Possibly through A54 -> 23 17 15 16 A23 17 (rule 702). That yields something like "23 17 15 16" then A23 then 17. A23 expands to (1) or A24 A23 (i.e., 1 followed by optional chain of A24). A24 yields 18,19,20. So A23 can generate a sequence of 18/19/20 followed by 1? Actually A23 -> 1 (or A24 A23). So A24 -> 18,19,20. So A24 A23 can produce "18 1" sequence (if A24 selects 18) and then A23 yields 1. Or we can generate longer sequences repeated. Let's think later.

So it's plausible that after A9 we could produce "23 17 15 16" then A23 (generates a sequence of 18,19,20 and maybe 1), then "17"? The target after "28 27" is "23 17 15 16 19 20 18 1 14 21 22 17". However note ordering: after "23", target has "17", then "15", "16". Wait target is "23 17 15 16 19 20 18 1 14 21 22 17". Sequence after 23: 17,15,16,19,20,18,1,14,21,22,17

But rule 702 yields "23 17 15 16 A23 17". That's "23 17 15 16", then A23, then "17". That matches exactly: we have after 23, we need "17 15 16", then something from A23 (maybe "19 20 18 1" etc), then "17". However note ordering of 19 20 18 differs: target has "19 20 18". In A23, we can generate a sequence of "A24 A23". A24 can be 18,19,20. Might generate 19 then 20 then 18? Possibly: A23 -> A24 A23 -> (19) + (A24 A23) maybe produce 19 20 18 and eventually 1. Let's examine how to produce "19 20 18 1": It could be A23 => A24 A23 (select 19), then A23 => A24 A23 (select 20), then A23 => A24 A23 (select 18), then A23 => 1 (terminate). Then we'd have sequence 19 20 18 1. So indeed A23 can produce 19 20 18 1 exactly. So the pattern matches: after "23 17 15 16", we can have A23 generate "19 20 18 1", and then the trailing "17" from rule 702 matches the final 17 after that.

Thus we have a part that yields "23 17 15 16 19 20 18 1 17". However target also includes "14 21 22" between the first 1 and the last 17? Wait actual target after "19 20 18 1" is "14 21 22 17". Let's check: The target part after "23 17 15 16 19 20 18 1 ..." is "14 21 22 17". Indeed after the "1" there is "14 21 22 17". So we need to generate that "14 21 22" before the final 17. But the rule 702 yields "A23 17" where we have after A23's generated sequence, we get a terminal 17. Since A23's possible expansions include 1 at the end. This gives a "1" before the final 17. But we still need "14 21 22" before the final 17. So we need to generate "14 21 22" somewhere, maybe via another production chained after A23 but before the ending 17. However the rule 702 fixed order: ... A23 (some sequence) 17. So all symbols after A23 are just the final 17. If we need "14 21 22" before the final 17, we could insert terminals "14 21 22" after A23, before the final 17, in rule 702 by insertion. That would be three terminal insertions.

But maybe we can generate "14 21 22" via A23 expansions? Let's examine A23 again: It can only produce the terminals 1 (direct), or A24 A23. A24 can produce 18,19,20 (three options). So no 14,21,22. So they cannot be generated via A23.

Thus we need to insert terminals "14 21 22" into some production after A23 but before the final 17.

Alternatively, maybe we could use a different production to generate those numbers, for example rule 474: A37 -> 23 A23 14 21 22. This yields exactly the subsequence we need: 23 A23 14 21 22. That yields "23 ..." plus after that "14 21 22". Then we could generate 17 afterwards via some other production. Actually rule 474 yields "23 A23 14 21 22". There's no trailing 17. But we could combine with a subsequent production that yields "17". For example, maybe after A37 we have some nonterminal that can produce 17. Let's examine A37 productions:

- rule_474: A37 -> 23 A23 14 21 22
- rule_475: A37 -> 23 A23 17
- rule_476: A37 -> A30 A36
- rule_477: A37 -> A31 A37
- rule_478: A37 -> A37 A35
- rule_479: A37 -> A42 A38

Thus we might use a combination: A37 -> 23 A23 14 21 22 and then incorporate some other production that yields "17". But how will we attach that 17 after the 22? Possibly via A37->A31 A37? Recursively we could generate more after 22, but that would place the 23... within A37's children not after.

Better consider using rule 474 for initial "23" and then after that we could have A23 produce 19 20 18 1, then we need "14 21 22 17". However rule 474 already includes 14 21 22 after A23. So we could use that: "23 A23 14 21 22". Then need trailing 17. We can get 17 via a later production from e.g., A41->... A41->A41 A41... Not sure.

But we could also combine rule 475 which yields directly "23 A23 17". That missing 14 21 22. So we could use rule 474( is good) for 14 21 22, and then need a following "17". Perhaps we can get "17" via a following nonterminal in the overall grammar after A37, e.g., A4 -> A20 after producing A37. But we need to check.

The sequence after "23 17 15 16 19 20 18 1 14 21 22 17" is "9 7 10 11 5 3 4 1 6 8 2 7". That likely corresponds to production(s) generating the rest: e.g., A43 -> 9 10 11 A21 7, A60 -> 9 7 10 A21 8 7 etc. Also there is A44 with sub expansions maybe.

Thus we may need to combine multiple sub-strings: after the "23 ... 14 21 22 17", we could produce "9 7 10 11 ..." via some other chain.

Thus we could use productions to generate these final segments.

Now we have two candidate substructures: 
- The initial "30 27 13 29 ... 28 27" path uses A9 (with inserted 27). It yields "30 27 13 29 x28 27" where x = A25 expansions produce "24 26 25 1". Good.

- Then we need "12 31". 12 is from A66 -> 12. 31 missing.

We could generate "12 31" via a production that yields 12 then we insert 31 after, perhaps in the same rule or a different one. Where can we generate 12? A66 produces 12, but in our current derivation after A9 we are at some nonterminal chain: we started A2->A61 A3. A61 gave 1 1. A3 gave maybe A4 -> A5 -> ... or A4 -> ... etc.

Our current plan: Let A3 start with A4 -> A5 -> A9 A19 (since A5 -> A9 A19). That would generate A9 (the block with 30 series) then A19. So need A19 to produce the remaining part: after A9's block we still need "12 31 28 27 23 ...". Actually A9 gets us up to "... 28 27". After that we still have "23 ..." starting next. So we could have A5 -> A9 A19 where A9 yields the block, and A19 yields the rest: "12 31 28 27 23 ..." Wait but note A9 already yields "30 27 ... 28 27". That includes "28 27". So next token after A9 is the token after that, which is start of A19's derivation.

But target after that block is "23 17 15 16 ...". Actually there is "12 31" before that. In target we have "... 1 12 31 28 27 23 ..."? Let's re-evaluate target after first two 1's: Sequence:

Indices:
1:1
2:1
3:30
4:27
5:13
6:29
7:24
8:26
9:25
10:1
11:12
12:31
13:28
14:27
15:23
16:17
17:15
18:16
19:19
20:20
21:18
22:1
23:14
24:21
25:22
26:17
27:9
28:7
29:10
30:11
31:5
32:3
33:4
34:1
35:6
36:8
37:2
38:7

Now A5 -> A9 A19 yields A9 (positions up to 14) and A19 (positions 15-38). Wait does A9 produce up to position 14? Let's see: A9 -> 30 27 13 29 A25 28 27. A25 yields 24 26 25 1 (positions 7-10?). Actually the structure:

- A9 start -> terminals [30,27,13,29] then A25 -> sequence of numbers (maybe 24, 26, 25, 1) then terminal 28 then terminal 27.

Thus overall A9 will generate exactly tokens: 30,27,13,29, (24,26,25,1), 28,27. Let's compute token positions:

Positions:
3:30
4:27
5:13
6:29
7:24 (via A25)
8:26 (via A25)
9:25 (via A25)
10:1 (via A25)
11? Wait after A25 we have terminal 28 (this would be position 11). But target token at position 11 is 12. Actually token at position 11 in target is 12, not 28.

Examining target: after 1 we have 12 (pos 11). The A9 yields 28 at that point, so mismatch. We need to insert a different terminal maybe before 28? But there is a 28 at position 13 in target. Indeed target token at position 13 is 28. And token at pos 12 is 31. So we need to have "28" at pos13. A9 currently outputs "28" right after A25 (i.e., before 31). But we need 12 and 31 inserted before 28. So we could modify the rule to insert "12 31" before the terminal 28. Since we can insert terminals anywhere in a production. So we could change rule_210 to insert "12 31" before the "28". That would produce: 30 27 13 29 A25 12 31 28 27 (but we also need to keep the inserted 27 earlier after 30? Let's think.)

Wait we inserted a 27 after 30 earlier to match early 27. So current insertion plan: modify rule_210 to:

A9 -> 30 27 13 29 A25 12 31 28 27

But also maybe we need to insert "12" and "31" after A25, before 28.

But check the target sequence after A25: after "1" (pos 10) we have 12 (pos 11), 31 (pos 12), 28 (pos 13), 27 (pos 14). So indeed we need 12 31 before 28 27.

Thus we can insert 12 and 31 at appropriate points. However, we must also include 12 terminal (available) and 31 terminal (currently missing). So we need to insert two terminals (including the missing 31). We already added one insertion for 27 after 30. So total inserted terminals then would be 3: a 27, a 12, a 31.

But careful: rule_210 currently has structure: "30 13 29 A25 28 27". We changed to "30 27 13 29 A25 12 31 28 27". Let's count inserted terminals: we inserted 27 after 30; inserted 12 after A25; inserted 31 after 12 (or after A25). So 3 insertions.

Now we also need to ensure the "12" is produced by an actual terminal, which is allowed (terminal 12 is present). And "31" is also a terminal, just missing from grammar but we can insert directly; we don't need to produce 31 elsewhere.

Now after we have A9 generating tokens up to 27 (pos 14), the rest of target from pos 15 onward is "23 17 15 16 19 20 18 1 14 21 22 17 9 7 10 11 5 3 4 1 6 8 2 7". This we must generate via A19 (since after A9 we go to A19). Let's see if A19 can produce that sequence.

Check A19 productions:

- rule_257: A19 -> 1
- rule_258: A19 -> A7 A5
- rule_259: A19 -> A8 A19
- rule_260: A19 -> A14 A20
- rule_261: A19 -> A19 A18

Thus A19 can recursively produce many things.

Consider A19 -> A14 A20 (rule 260). A14 is a nonterminal that can generate many forms, including eventually producing 23 via rule 474? Wait A14 maybe does not directly produce 23; A14 productions: rule_231 (A14 -> 1), rule_232 (A14 -> A7 A13), rule_233 (A14 -> A8 A14), rule_234 (A14 -> A14 A12), rule_235 (A14 -> A19 A15), rule_236 (A14 -> 23 A23 22). You wrote: rule_236 is "A14 -> 23 A23 22". Actually that's missing 14? Wait rule 236: A14 -> 23 A23 22. So A14 can generate "23 A23 22". That would give 23 then something from A23 and then 22. But we need "23 17 15 16 ... 14 21 22 17". So A14's production includes 23 and 22, but missing 17 15 16 and the later sections.

Note that there is also rule_474 earlier in A37 which does "23 A23 14 21 22". That includes 14 and 21 as well, but not 17, etc.

Now A20 we described earlier; it can generate "9 7 10 A21 6 7" or other.

Thus A14 A20 might be promising to generate the whole suffix: A14 could generate "23 ... 14 21 22" and A20 could generate the final "9 7 10 ..." plus maybe preceding "17".

But we need to generate "23 17 15 16 ... 14 21 22 17 9 7 10 11 5 3 4 1 6 8 2 7". Let's see: A14 -> 23 A23 22 (rule 236) gives 23, then A23 (which expands to 1 or 18/19/20 combos), then 22. To incorporate the "14 21" between 1 and 22 we might need another production like A14 -> ??? Actually there is rule_474 with A37 for 23 A23 14 21 22.

But A14 does not produce the 14 21. A37 does. However A19 can produce A14 A20. So perhaps we can use A14 -> 23 A23 22, then produce '14 21' by something else before A20? Actually A20 starts with 9 7 10 etc., doesn't produce 14 21. So not.

Alternatively, A19 can go to A7 A5 (rule 258). A7 may produce complex sequence that might contain 14 21 etc. Let's examine A7 productions:

Rule 195: A7 -> 1

Rule 196: A7 -> A7 A6

Rule 197: A7 -> A8 A7

Rule 198: A7 -> A14 A11

Rule 199: A7 -> A19 A16

Thus A7 can produce A14 A11 (rule 198). Then A11 could produce sequences.

A11 productions:

- 216: A11 -> 1

- 217: A11 -> A10 A7

- 218: A11 -> A11 A6

- 219: A11 -> A12 A11

- 220: A11 -> A20 A16

Thus A11 can produce A10 A7, which can produce many.

But perhaps it's easier to use A19 -> A14 A20, and then insert needed terminals to fill gaps, like inserting "17 15 16 ... 14 21 22 17". However we have restrictions: only insert terminals, cannot reorder.

Consider the production A14 -> 23 A23 22 (rule 236). This yields "23" then whatever A23 yields (some sequence of 18/19/20/1) then "22". In target we need "23 17 15 16 19 20 18 1 14 21 22 17". The pattern is:

23
17 15 16
19 20 18
1
14 21
22
17

It seems we could attempt to use A14 -> 23 A23 22, and then insert the missing terminals "17 15 16" after the leading 23 and before A23; and also insert "14 21" before the final 22; and also insert final "17" after 22 (but we might get 17 from elsewhere). Or maybe we can use A23 to produce "19 20 18 1". Yes.

Thus with A14 -> 23 A23 22, we could insert:

- "17 15 16" after 23 (i.e., between 23 and A23)

- No need to insert "14 21" if we can get them elsewhere; we need them between the 1 and final 22: Actually after A23's sequence (which ends with 1, but might have 1 earlier?), we need "14 21" before 22. insertion after A23 but before 22 can include "14 21".

Thus final "17" after 22 could be inserted after 22 as well.

Thus by inserting terminals "17 15 16" after 23, "14 21" after A23, and "17" after 22, we can get the desired output.

Alternatively, if we can use other productions that already include some of these numbers, we could reduce insertions.

But note also that there is rule 474 (A37) that includes "23 A23 14 21 22". This includes 23 then A23 then 14 21 22. So that covers many of the needed tokens, but lacking the "17 15 16" after 23 before A23, and also the trailing "17". So we could use A37 (instead of A14) and maybe attach its trailing "17" via further productions.

Let's check A37 productions: rule_474 (A37 -> 23 A23 14 21 22) and rule_475 (A37 -> 23 A23 17). Also rule 476, etc. So A37 can produce the pattern "23 A23 14 21 22". Good. We could then after A37 produce something that yields "17 15 16 ... 17"? But A37 does not produce these preceding numbers "17 15 16". We still need "17 15 16" after 23, but before A23? Actually the rule 474 already has 23 A23 ... so can't put items between 23 and A23 unless we insert.

But we could also use rule 475: "23 A23 17". That yields 23 then A23 then 17. That covers the final 17 but missing "14 21 22" earlier and missing "17 15 16". However we could combine multiple A37 applications perhaps using recursion via A37 -> A31 A37 etc. Could we embed the 14 21 22 before the final 17 using recursion? For example, we could have first A37 produce "23 A23 17". Then after that (maybe inside recursion, the next A37 could add "14 21 22" via some other path? But A37 recursion is via productions: 476: A37 -> A30 A36; 477: A37 -> A31 A37; 478: A37 -> A37 A35; 479: A37 -> A42 A38. Those produce huge things but not directly the terminal sequences.

Probably it's simpler to use A14 -> 23 A23 22 and insert necessary numbers. We need to calculate total insertions: If we use A14, we need to insert:

- after 23: 17 15 16 (three insertions)
- after A23: 14 21 (two insertions)
- after 22: 17 (one insertion)

Total 6 terminals inserted.

But if we use A37 rule 474 (23 A23 14 21 22) we need to insert:

- after 23: 17 15 16 (3 insertions)
- after 22: 17 (1 insertion)

Total 4 insertions.

But the final sequence after 22 also includes the "17" that we would need. So 4 insertions.

Now we also need to ensure the "14 21" from rule 474 are already present, so we don't need to insert them. Good.

Now we also need to generate the remaining suffix after that: after the final 17, we need to generate "9 7 10 11 5 3 4 1 6 8 2 7". This could be produced by A20? Let's verify.

A20 can produce "9 7 10 A21 6 7" (rule 263). That yields 9,7,10, some nonterminal A21,6,7. A21 -> 1 | A22 A21 (rule 268*). A22 -> 3,4,5 (so can generate 3,4,5 via sequence? Actually if A21 -> A22 A21 we can choose numbers 3,4,5 and then maybe end with 1. That's promising. So A20 produces "9 7 10" then A21 yields some sequence of 3,4,5 and eventually 1, then "6 7". Let's compute typical A21 derivation: A21 -> A22 A21 -> maybe A22->3; then A21 continues -> A22 A21 -> maybe -> 4 then continue -> A22 A21 -> maybe -> 5 then continue -> maybe -> 1 (via A21->1). So the sequence would be 3 4 5 1. But note order: A21 -> A22 A21 (first A22 then A21). So if we repeatedly choose A22 as 3 then again 4 etc., the order would be 3 4 5 ... followed by 1 at the end. The target after "9 7 10" is "11 5 3 4 1 6 8 2 7". Wait we have "11 5 3 4 1 6 8 2 7". So we need "11" before the 5. Our A20 rule 263 only yields up to A21, not 11.

But there is also rule 506: A43 -> 9 10 11 A21 7, which yields 9,10,11, then A21, then 7. That yields "9 10 11 ... 7". Not exactly 9 7 10 11 as per target. Actually target begins "9 7 10 11". But may be generated by a series of productions: we could produce "9 7 10" via A20, then after that "11" might be generated by something else like A41->... but not. But order may differ.

Alternatively, we could use A20 -> A10 A19 etc. We need to see if we can generate "9 7 10 11 5 3 4 1 6 8 2 7" with minimal insertions. Perhaps better to combine A20's "9 7 10 A21 6 7" and add "11" after 10 but before A21 (insert terminal 11). That would give "9 7 10 11 A21 6 7". Then A21 could generate "5 3 4 1". But we would need "5" before "3". A21 can only generate A22's which produce 3,4,5 but not 5 before 3. However we could insert terminals to reorder? We cannot reorder tokens, only insert. So we need to ensure that A21 yields "5 3 4 1". But A21's grammar yields tokens of A22 (which can be 3,4,5 individually). The order depends on sequence of A22 expansions; we can choose as per our need: we can first expand A21 -> A22 A21 where A22 produces 5 (choose rule 273? Actually A22 -> 5). Then A21 -> A22 A21 where A22 produce 3, etc. But we need "5 3 4 1". Yes we could have A21 produce 5 then 3 then 4 then 1. Let's check: A21 -> A22 A21 (first A22 could be 5). Then recursively produce A21 -> A22 A21 again (first now maybe 3). Then again (now 4). Finally A21 -> 1. So the sequence would be 5 3 4 1. Works!

Thus we can generate "5 3 4 1" using A21's recursion. Good.

After that we have "6 8 2 7". In the A20 rule we have after A21 the terminals "6 7". So we get "6 ... 7". We need to insert "8 2" before the final 7 maybe after the 6. Actually after "6", we need a "8 2" before "7"? No, the target after "1" is "6 8 2 7". So we can get "6" from rule, then insert "8 2" before the "7" terminal: note rule 263 ends with "... 6 7". So we can insert "8 2" between "6" and "7". That's two inserted terminals.

Thus, using rule 263, we need to insert:

- "11" after "10": actually we need "9 7 10 11 ..." The rule gives "9 7 10 A21 6 7". So after "10" we have A21. We want "11" before A21. We can insert "11" after "10". That's one insertion.

- "8 2" after "6" as described: two insertions.

Total insertions for final suffix = 3 (one for 11, two for 8,2). Weighted earlier recognized we also need to insert a terminal 27 after 30, and 12 and 31 after A25: that's 3. plus we need insert "17 15 16" after 23, "17" after 22 (if using rule 474). That's maybe 4 (3+1 = 4). Wait we previously counted using A14 to insert 3+2+1=6. Using A37 reduces to 4 (3+1). Let's verify: If we use A37 rule 474 (23 A23 14 21 22). Then we need to insert "17 15 16" after 23 (3 insert), and "17" after 22 (1 insert). So total 4.

Thus total insertions would be: (in rule_210) we inserted three terminals (27 after 30, 12 after A25, 31 after 12). That's 3.

Plus (in rule_474) we inserted four terminals (17,15,16 after 23 ; 17 after 22) = 4.

Plus (in rule_263) we inserted three terminals (11 after 10; 8 after 6; 2 after 8) = 3 insertions.

Total insertions = 3+4+3 = 10.

But we also might need to fix ordering of "28 27" after insertion of 12 and 31 - we inserted them before 28; we need to ensure "28 27" remain at end segment. That's okay.

But do we need any other modifications? Perhaps target includes "14 21" before "22" which are already present in rule 474 (23 A23 14 21 22). Good.

Now need to verify that using A14 vs A37 - we choose A37. But where does A37 appear? We need A14 (or A37) inside the remainder derived from A19. In A19 we have the options: we could use A14 A20 (rule 260). That would use A14 (which for our needed sequence we can substitute with A37? Not directly. A14 and A37 are different nonterminals. A14 can produce 23 A23 22 (needs insertions). A37 directly yields the needed sequence but we need to get A37 from somewhere. Let's see if A14 can be replaced with something else: maybe A19 -> A14 A20 can be used but we can use A14's production 236 (23 A23 22) and then we need to insert "14 21" and other missing terminals. That's more insertions (total 6). But maybe A19 can produce A14 A20 where A14 can produce "23" and "14 21 22"? Actually rule 236 (A14 -> 23 A23 22) does not give 14 21; but there is also A14 -> ??? We saw A14 -> 23 A23 22 only. But A14 also has other productions: "A14 -> A14 A12" etc; "A14 -> A19 A15" etc. But none directly yields "14 21". So better to use A37.

But can we get A37 from A14? Not directly unless we make A14 -> A37 (by insertion? Can't replace nonterminals; only insert terminals. So cannot.

Thus our path using A19 -> A14 A20 (rule 260) may need insertions: perhaps we could use A14 -> 23 A23 22 and then before 22, we can insert "14 21". That would be two insertions (14 and 21). And then after 22 we insert "17". Also after 23 we need "17 15 16". That's three insertions. So A14 path would need 6 insertions.

But we also need to produce "14 21" via separate insertion or not. So total for A14 path 6 insertions. That's higher than A37 path 4 (if we get A37). However we need to get from A19 to A37. A19 doesn't have a rule to produce A37 directly. But we could use A19 -> A7 A5 (rule 258) then further expansions could produce A37. For instance, A7 can give A14 A11 (rule 198), then A11 might generate A20 (rule 220: A11 -> A20 A16). That could bring A20 into the mix after A14. But A14 itself can produce the needed 23 A23 14 21 22 via rule 236? Actually A14 has rule 236: "A14 -> 23 A23 22". Not enough.

But if we use A7: A7 -> A14 A11 (198). Then A11 -> A20 A16 (220). So the sequence becomes: A14 A20 A16. Then A14 could produce 23 ...; A20 would produce the terminal suffix; and A16 etc. However we also need "17 15 16" after 23 (the numbers before A23). Using A14 (236) we have 23 A23 22. Then we can insert "17 15 16" after 23 (3 terminals). After A23 we also need "14 21" before 22 (2 terminals). However, we also need trailing "17" after 22. That's 1 insertion. That's total 6 inserted terminals as earlier.

But we also have A16 present after A20 (since A11->A20 A16). A16 can produce many numbers, e.g., rule 242: A16->1, also other expansions. Might produce needed "17"? Actually A16 can produce something else, maybe we could get the final "17" without insertion via A16 expansions. Let's check A16 productions further:

- rule_242: A16 -> 1
- rule_243: A16 -> A15 A11
- rule_244: A16 -> A16 A6
- rule_245: A16 -> A17 A7
- rule_246: A16 -> A18 A16

Thus A16 can generate other sequences. It may produce terminal "17"? Let's see if any production yields 17 directly: Nothing shows 17 as a terminal in these rules. But there is rule 801? Actually A67 may produce 1 only; no 17. So we may need insertion for that final 17 anyway.

Thus the A14 replacement path (via A7->A14 A11) still needs insertion for 17 after 22.

But the A37 route could be shorter: we need to get A37 somewhere without many insertions.

Can we get A37 from nonterminals in A19? A19 does not directly produce A37, but we could use A19 -> A14 A20 as earlier. However we can use A14 to generate A37 using insertion? Not allowed.

Alternatively, A19 could be expanded via recursion: A19 -> A19 A18 (261). That could produce A19 then A18 (which yields 1). If we use A19 recursively, we could produce A19 multiple times. So perhaps the first A19 we generate (via recursion) could produce A14 A20 (i.e., A19 -> A14 A20 using rule 260?), but we need to get to A14 (via rule 260) not A19. Actually rule 260 is A19 -> A14 A20. So we can directly have A19 produce A14 A20. Good, we already have that. So we can produce A14 A20.

Now A14 (as before) can produce "23 A23 22" (rule 236). We need "14 21" between the A23 part and 22? Wait rule 236 currently is "A14 -> 23 A23 22". It does not have "14 21". However there is rule 236 we could consider inserting "14 21" after A23, before 22. That's two insertions. Then we still need "17 15 16" after 23, before A23, and a trailing "17" after 22 maybe via insertion. So total insertion for A14 path is 3+2+1=6. However we might also generate "14 21" using some other production inserted after A23 via A14 -> A14 A12 or similar? Let's examine other productions for A14 to see if they can produce the needed "14 21". For example:

- rule 232: A14 -> A7 A13
- rule 233: A14 -> A8 A14
- rule 234: A14 -> A14 A12
- rule 235: A14 -> A19 A15

Maybe we could use A14 -> A7 A13 to produce 14 21. Does A7 produce "14"? Possibly via rule 198: A7 -> A14 A11, but that is recursive.

But we could also try using A14 -> A19 A15: then A15 could generate "14"? Let's see A15 productions: rule 237: A15 -> 1; rule 238: A15 -> A15 A12 ; rule 239: A15 -> A16 A13 ; rule 240: A15 -> A17 A14 ; rule 241: A15 -> A18 A15. None produce terminal 14.

But maybe A12 produces 14? Let's examine A12 productions: rule 221: A12 -> 1; rule 222: A12 -> A10 A14; rule 223: A12 -> A11 A13; rule 224: A12 -> A12 A12; rule 225: A12 -> A20 A15. So A12 can include A14 which may produce 14? Not directly.

Thus better to accept insertion for "14 21".

Thus it appears the A14 path is indeed more insertions than the A37 path.

But to use A37 we need to get A37 from A14 maybe? Actually no, we can produce A37 directly because A19 -> A14 A20 (rule260) uses A14, not A37. However, perhaps we can produce A37 using different path: e.g., A19 -> A7 A5 (rule 258) -> A7 -> A14 A11 (198) -> A11 -> A20 A16 (220). That yields A14 A20 A16. The A14 part can produce needed 23 A23 14 21 22 via rule 236 plus required insertions (but that's same). But we could also modify A14's production to be A37? Not allowed.

But there is rule 235: A14 -> A19 A15 (so could produce another A19). Not helpful.

An alternative: use A19 -> A19 A18 (261). This could create repeated A19 expansions. Maybe we could do A19 -> A19 A18 (unordered) to generate something after the main A14 A20? However A18 is just 1, presumably generating a trailing 1.

Thus A37 may not be reachable directly.

However maybe we can produce A37 via top-level A3, e.g., choose A3 -> A27 (rule_39). Since A27 can produce A41, A42, A43 etc., maybe we can generate a different order: after A9's block (we might generate via A5->A9 A19, but we could also generate a different path remainder that includes A37. Let's verify.

A27 productions:

- Many of them: A27 -> A28 ... (lots)
- A27 -> A41 (starting from rule_319)
- A27 -> A42 (rule_355)
- A27 -> A43 (rule_391)
Also early A27 -> A28 (lots). So A27 can produce A41, A42, A43. A41 can produce terminals as lots. A43 can produce "9 10 11 A21 7". A41 may also produce something that yields the 14 21 22 etc? Let's examine A41 productions:

- rule_495: A41 -> 1
- rule_496: A41 -> A38 A43
- rule_497: A41 -> A39 A28
- rule_498: A41 -> A40 A42
- rule_499: A41 -> A41 A41

Thus A41 can produce A38 A43, which includes A43 (the 9 10 11 A21 7) plus prefix A38 (which may generate 14,21 etc?). Check A38 productions: rule_474 and 475 etc (they produce sequences with terminals 23 and 14 etc). Actually rule_474 is "A37 -> 23 A23 14 21 22". That's A37, not A38. Wait we need to check: A38 productions? In the list we have:

- rule_474 is for A37. A38 productions are:

From earlier, we saw:

rule_474? Actually we need to locate A38 definitions. Let's search: we have rule_474 is A37; rule_475 is A37; rule_474's neighbors for A38: rule_480: A38 -> 1
rule_481: A38 -> A38 A35
rule_482: A38 -> A39 A36
rule_483: A38 -> A40 A37
rule_484: A38 -> A41 A38

Thus A38 can produce A41 A38 (recursive). So A41 A38 could produce A41 then again A38 etc.

Thus A41 -> A38 A43 could eventually produce A41 (via recursion) and then A43.

Thus perhaps we can assemble a sequence using A27 -> A41, then A41 uses A38 A43 producing a prefix with some other numbers and then the final 9 10 11 etc.

But the numbers we need after the 23 fill include "9 7 10 11 5 3 4 1 6 8 2 7". A41 may produce that.

But we can also target using A43 directly: A43 -> 9 10 11 A21 7 (not 9 7 10). That is close but missing the 7 after 9 and before 10. We'll need an insertion of "7" after 9.

But note the target expects "9 7 10 11". So using A43 base we have "9 10 11". Insert "7" between 9 and 10 yields "9 7 10 11". That's one insertion.

Alternatively, we could use A20 rule 263 "9 7 10 A21 6 7". That's "9 7 10 ... 6 7". This gives 9,7,10 (good). After A21 we need "11". So we could insert "11" after A21 before 6. That's one insertion. Then the rest "6 7" we need "6 8 2 7". So we would need to insert "8 2" between 6 and 7: two insertions. So total three insertions we previously counted.

If we use A43 path we need insertion "7" after 9; plus the rest we need to generate "5 3 4 1 6 8 2 7". Using A43's A21 yields a sequence like "5 3 4 1". So after that we only need "6 8 2 7". Could we get "6 8 2 7" via inserting "6 8 2" before 7? Actually rule 506 ends with "A21 7". So after A21 we get terminal 7. So we need to insert "6 8 2" before the final 7. That's three insertions. So total A43 method: "7" after 9 (1), "6 8 2" before 7 (3) → total 4 insertions. So A43 method results in 4 insertions.

We previously computed A20 method as 3 insertions (11 after 10, 8 and 2 after 6). Wait we also need final "9 7 10 11". A20 gives 9,7,10, then we need 11 inserted = 1. Good.

Total with A20 method: insertion of 11 after 10 (1), insertion of 8 and 2 before final 7 (2) = 3. So A20 method uses 3 insertions, which is fewer than A43's 4.

Thus best to use A20 to generate suffix.

Now we need A20 to be reachable from where? A19 -> A14 A20 (or via other). So we need A20 via A14 A20 structure: A14 will generate the earlier part "23 ... 14 21 22", and A20 will generate suffix "9 7 10 11 5 3 4 1 6 8 2 7". Wait A20's direct productions: rule 263 "9 7 10 A21 6 7". That yields 9,7,10, then A21 (which can generate 5 3 4 1), then 6, (then 7). So overall we get "9 7 10 (5,3,4,1) 6 7". With insertions, we get "9 7 10 11" (by inserting 11 after 10) and "9 7 10 11 5 3 4 1 6 8 2 7" (by inserting 8,2 before 7). That's perfect. So A20 works.

Thus we need the following insertions:

- In rule_263 (A20 -> 9 7 10 A21 6 7): Insert "11" after "10". Possibly we need to consider that there is already "9 7 10" then A21. Insert 11 before A21.

- Insert "8 2" after "6". Actually rule_263 ends with "6 7". Insert "8 2" between "6" and "7".

Thus three insertions for suffix.

Now we need the earlier part using A14. Let's see A14's production options. We want a production that yields "23 A23 14 21 22" with minimal insertions. The rule 236: A14 -> 23 A23 22. Then we need to insert "14 21" after A23, before 22: two insertions. Also we need "17 15 16" after 23: three insertions. After 22 we need trailing "17": one insertion. That's total 6 (as earlier). But we could consider using rule 236 plus insertion of "14 21" plus others. That's 6 total.

But maybe there is a better A14 production that includes some of these numbers already. Let's search for A14 productions again:

- A14 -> 1
- A14 -> A7 A13
- A14 -> A8 A14
- A14 -> A14 A12
- A14 -> A19 A15
- A14 -> 23 A23 22

Only the last one includes 23 and 22. None include 14 or 21. So we need at least insert 14 and 21.

Now maybe we can avoid using A14 itself; perhaps use A7->A14 A11 as part of A19 to get A14 then later A20. But still we'd need A14.

Thus, using A14's part requires 6 insertions.

What about A19 -> A7 A5 (rule 258). Maybe we can use A7's production to generate the required pattern directly without using A14. Let's examine A7.

A7 productions:

- A7 ->1
- A7 -> A7 A6
- A7 -> A8 A7
- A7 -> A14 A11
- A7 -> A19 A16

Potentially we could use A7 -> A14 A11. So A7 yields A14 then A11. Then A11 can produce something. In A11's productions: A11 -> A20 A16 (rule 220) among others. So A7 -> A14 A11; A11 -> A20 A16. So overall A7 can generate A14 A20 A16. This yields A14 (for earlier) then A20 (suffix) then A16 (maybe generate trailing 17). That's exactly similar to using A19 the other way. So still have A14.

Thus overall if using A14 we still need 6 insertions for that segment.

Now maybe we can use A37 in place of A14 by directly getting A37 via A7->something? A7 -> A14 A11, then A11 -> maybe A20 A16; but maybe we could use A11 recursion to get A37? Not directly. But A11 can produce A10 A7 (217) or other; could possibly produce A37 via deeper expansions. Let's examine A10 productions: A10 -> ... maybe can generate A37? Let's see:

- A10 -> A12 A10
- A10 -> A20 A17
- A10 -> A11 A9
- A10 -> A10 A8
- A10 -> 1

None produce A37.

A11 can produce A10 A7 (217); further expansions of A10 might produce something else. But no direct A37.

Thus likely path includes A14.

Now we think about total insertions might be 10 as earlier (3 from A9 rule, 6 from A14 rule, 3 from A20 rule) = 12? Actually A14 insertion is 6, A9 3, A20 3 total 12. Wait earlier I counted 10 (A14 using A37 with 4). But using A14 path we have 6 for A14, 3 for A9, 3 for A20 = 12. That's more.

But perhaps we can reduce insertion count elsewhere. Maybe we can combine some of those insertions.

Let's analyse the A9 modification: we inserted 27 after 30, plus 12 and 31 after A25.

But we could also insert 27 after 30 with rule_210, and we need 12 and 31 after A25. That's 3.

But perhaps we could use a different production that yields 30 27 and also includes 12 and 31 elsewhere, reducing insertions. E.g., maybe we could use A32 -> 30 27 A25 (rule 450) and then after A25, we could generate "12 31 28 27" via further expansions rather than inserting 12 and 31. Could A25 generate "24 26 25 1 12 31 28 27"? It currently only expands to A26 A25 or 1. Could we insert "12 31 28 27" after A25 via later expansions? Actually after A25 we may get 28 27 from the rest. In rule 450 there is no 28 27; we would need to generate those as well. The rule 450 gives 30 27 A25 with no 28 or 27 after. So we need to insert 13 29 ... etc. That seems not good.

Alternatively, we could use rule_210 plus insert 27 after 30 and maybe generate 12 31 28 27 using A... alternative expansions? Let's check A25's expansions: A25 -> A26 A25, can keep generating numbers 24,25,26 in any order, then terminates with 1. So cannot generate 12,31,28,27. That part must be inserted.

Thus we cannot reduce insertion count there.

Now for the suffix, we need to produce "9 7 10 11 5 3 4 1 6 8 2 7". In A20 we inserted 11 after 10, 8,2 after 6 = 3 insertions.

Can we reduce? Perhaps we can use A43 (9 10 11 A21 7) and insert 7 after 9 (to get 9 7 10...). And also need to generate "5 3 4 1" and then "6 8 2 7". But after that we have "7" already, we need to insert "6 8 2" before the final 7 (that's 3 insertions). So total for A43 path = 4.

Could we use a combination: use A20 (9 7 10 ...) and A43 (9 10 11 ...) to share some? Probably not.

Thus A20 path seems minimal (3 insertions). But maybe we can modify A20 rule_263 to insert two terminals (11 after 10, and 8 and 2 after 6). Actually we need two insertions after 6 (8 and 2) and one after 10 (11). Yes.

Now consider the A14 part: maybe there is a production that gives 23 A23 14 21 22 directly; but that's A37, not A14. But maybe we can connect A14 to A37 via left recursion: for example, A14 -> A19 A15 and A15 could produce A37? No. A15 does not produce A37.

But maybe we can produce A37 via A20 or other nonterminals after A14? The target after 22 is "17". Could that be generated by A20's expansion (like A20 possible includes 17 as part of A21 expansions maybe)? Actually A21 generates 3,4,5 and then 1; not 17. So not.

Thus likely we need to insert "17" after 22 anyway.

If we could generate the 17 directly via a nonterminal that appears after 22 (like A15, A16 etc.) we could avoid insertion. Let's see: After A14's production we have 22, then possibly some other nonterminal via recursion? For A14 -> 23 A23 22, it's just terminals. So no nonterminal after 22.

But A14 could be used in a larger context where after A14 we have something else that generates 17, e.g., A14 is followed by some other nonterminal like A20, which would generate 17 as part of its own prefix? But A20 starts with "9". Not 17. So not.

But we could use A14 in a different rule: A7 -> A14 A11 (198). So after A14, we can have A11, which can generate a 17 maybe. Look at A11: possibilities: A11 -> A10 A7 (217) etc. Could any of them generate "17"? Not directly; but expansions later could produce 17 as part of sequences; for example A7 can eventually produce 17? Let's see if any nonterminal can generate "17". 17 is a terminal that appears in many productions: In many rules (like A53 etc) there is "23 ... 17". But where does a "17" appear as terminal alone? Only in rules which have "A21 -> A22 A21" etc not. But there are rules like "A66 -> 1"? Actually no.

Let's search: Terminal "17" appears in many productions: e.g., rule 795? Actually not, we saw "A68 -> 17" (rule 802: A68 -> 17). Also A68 also -> 22. So A68 can generate 17. So a nonterminal A68 can directly produce terminal 17. So we could have after 22 a nonterminal that expands to A68 -> 17, without insertion. This would require a production that contains a nonterminal after 22 (instead of just terminal). Can we modify a rule to insert a nonterminal? No, we cannot change nonterminals, only insert terminals. So we cannot replace a terminal with a nonterminal. However, we could maybe incorporate a path where after 22 we have more symbols via an existing production that yields "22" not as a terminal but as part of a longer production that adds a nonterminal after it. But we would need to insert nonterminals, not allowed. So we cannot avoid insertion for the trailing 17 after 22 unless we already have a production that includes both 22 and something else after, like "22 X" where X is nonterminal. Do we have any production that has "22" followed by something? Let's examine all productions for "22". In rule 474: A37 -> 23 A23 14 21 22. That's terminal 22 at end. No trailing nonterminal.

A22 -> 5 maybe? Not.

Also rule 475 (A37 -> 23 A23 17) not 22.

Rule 702 says "23 17 15 16 A23 17". No 22.

Thus no existing production has "22" followed by something. So we need to insert "17" after "22" as a terminal.

Thus insertion is necessary.

Now consider insertion of "14 21". Instead of inserting these two, maybe we could restructure using a different production that includes "14 21" directly. For example, A37 includes "14 21". But we can't use A37 directly since A14 cannot produce it. However maybe we can use A14 -> A7 A13 (rule 232) and A7->... produce "23 A23 14 21 22". Could be possible: For instance, we could have A7 -> A14 A11 (198) and then A14 could produce the 23 part etc. That's cyclic.

Alternatively, maybe we can use A4 -> A19 after A9 (i.e., A5 -> A9 A19); A19 could produce A7 A5, which could generate A14 etc. Might be more insertion.

Thus the simplest approach is to use A14's '23 A23 22' and insert '14 21' after A23 then '17' after 22. That's 3 insertions after 23 and 2 after A23? That's total 3+2+1=6? Wait we inserted 3 after 23: "17 15 16", then after A23 we need 14 21 (2) plus after 22 we need 17 (1) = total 6.

Thus overall total insertions = 3 (A9) + 6 (A14) + 3 (A20) = 12. Could we reduce some from A9 path? Let's revisit A9 modifications:

We inserted 27 after 30. Is that necessary? Let's see if we could perhaps use a different production for the early part that yields 30 27 without insertion. For example, rule 450: A32 -> 30 27 A25. Could we use A5 -> A6 A5 or something to incorporate A32? Actually A5 does not produce A32. But maybe A4 could produce A2? Not. Let's see which nonterminal can produce something that yields A32. A4 can produce A5, A18, A19, A20 only. No A32.

But A3 also can produce A44 (rule 40), A27 (rule 39), A4 (rule 38). So A27 might eventually produce A32? Let's examine A27 expansions: It can produce A28 (lots), A41, A42, A43. There's no direct A32. So no.

Thus A32 appears only reachable via what? Let's search for A32 as RHS: rule 383? Actually rule 381? Not. Let's see: I recall earlier: rule 381? Let's search: The list shows "A32 -> A28 A40", "A32 -> A29 A32", "A32 -> 30 27 A25", "A32 -> A32 A31", "A32 -> A36 A33". So to derive A32, we need a nonterminal that expands to A32, maybe via A28 or A30 or A33 etc. Which nonterminal can produce A32? A28 can produce A28 A41, A29 A28, A32 A42, A36 A43. So A28 can produce A32 (rule 430: A28 -> A32 A42). So A28 expands to A32 and A42. A28 appears as RHS in many rules: e.g., A4 -> A5 (no), A4 -> A18, A4 -> A19, A4 -> A20; not there. A27 -> A28 (many). So A27 can generate A28. So there is a chain: A3 -> A27; A27 -> A28; A28 -> A32 A42; A32 -> 30 27 A25 (or 30 27 A25). This would produce "30 27 ..." without needing to insert 27 after 30 as we used earlier. Good!

Thus we can generate "30 27 ..." using A27 pathway: A3 -> A27 (rule_39), then A27 -> A28 (multiple rules), then A28 -> A32 A42 (rule_430). Then A32 -> 30 27 A25 (rule_450). This yields exactly "30 27 A25". Then A42 can generate something after that which we need to be maybe "28 27"? Let's see what A42 expands to.

A42 productions:

- rule_500: A42 -> 1
- rule_501: A42 -> A30 A28
- rule_502: A42 -> A31 A42
- rule_503: A42 -> A37 A43
- rule_504: A42 -> A42 A41

Thus A42 could produce something that yields "28 27"? Possibly via A30 A28 etc. A30 can expand to various forms; maybe A30 can generate "28 27". Let's look at A30 productions:

- rule_437: A30 -> 1
- rule_438: A30 -> A30 A29
- rule_439: A30 -> A31 A30
- rule_440: A30 -> A37 A34
- rule_441: A30 -> A42 A39

No direct terminals for 28 or 27.

But perhaps A28 can produce A41 or A41 -> ... But 28 and 27 as terminals appear only as literals (i.e., the integer 28 and 27). Are there productions that have 28 as terminal? Yes, rule 210 includes terminal 28, rule 450 does not. The only occurrences of literal 28 appear in rule 210 (terminal 28). Also possibly elsewhere? Let's search manually: scanning list: rule_210 includes 28; also rule 450 doesn't. Possibly also rule 336? No. So literal 28 appears only in rule_210. Similarly, literal 27 appears in many productions: rule 210 includes 27, rule_450 includes 27 after 30; rule_474 includes not 27. There may be other productions with 27? Many nonterminals have numeric 27 (nonterminal A27) but not terminal.

Thus the only way to produce terminal 28 is via rule_210 (A9). So our earlier plan using A9 is necessary to generate terminal 28 (and maybe also the terminal 28 that appears after A25). So we must use A9 to get 28.

So our approach using A9 to produce the block that includes 28 is good.

Now for the early 27 after 30 we inserted; maybe we can avoid that insertion by using a different route to get 27 earlier. Could the 27 after 30 be generated via A9's production if we can interleave something else? Since rule 210's 30 is followed by 13, not 27, we need an extra 27 after 30. Perhaps we can generate that 27 using A23? Actually A23 produces terminals 1 or A24 A23 (which produce 18/19/20). Not 27.

Thus we need to insert 27 after 30. However perhaps we can get the 27 from the A9's trailing 27, but we need it earlier. We could perhaps shift order by using A5 -> A13 A20 (rule 189) maybe to generate 30 27 later? Not.

Thus likely we need to insert at least one 27 after 30.

Now for 12 and 31 insertions. Could we generate 12 via a nonterminal like A66? But A66 is not used in the A9 path. After A25 we could have a nonterminal that expands to A66 then A31 etc. However A9's production is fixed: '30 13 29 A25 28 27' . The only place we can insert terminals is after A25 we inserted 12 and 31. Could we insert a nonterminal after A25 to generate 12 and 31? No, only terminals can be inserted, not nonterminal. So we cannot avoid those insertions.

But perhaps we could generate 12 and 31 via expansions of A25 before we get to terminal 28; For example, we could modify A25's production to include new nonterminals? Not allowed (only insert terminals). So can't.

Thus we need insert 12 and 31.

Thus the 3 insertions for the initial block appear necessary.

Now for the middle part (A14). Maybe we can avoid insertion of "14 21" by using A37 instead. But we need A14? Let's see if we can get "23 A23 14 21 22" using a different nonterminal that appears after A5's block. Maybe we can generate that segment via A19 -> A7 A5 etc. But we need the 23 14 21 22 sequence. Let's examine if any nonterminal other than A14 can yield that sequence with fewer insertions. The rule 474 (A37 -> 23 A23 14 21 22) is the perfect match for that contiguous segment. Could we have A19 produce A37 A20 instead of A14 A20? Let's see if A19 can produce A37. A19 productions: 1, A7 A5, A8 A19, A14 A20, A19 A18. None produce A37 directly.

But maybe using recursion: A19 -> A7 A5, and A7 can produce A14 A11, as we saw. Then A14 can generate maybe 23 A23 22. Then A11 can produce A20 A16 (via rule 220), giving the suffix. So the segment would become A14 A20 plus more. But to get "14 21", we still need to insert.

But perhaps we could generate "23 A23 14 21 22" by using A14 -> 23 A23 22 (rule 236) and then having A20 generate "14 21" before its own prefix? No, 14 and 21 would appear before 23? Not.

Thus we may need to accept the 6 insertions for middle segment.

However maybe we can get 14 and 21 from A20 or A21 expansions? A20 doesn't give 14 or 21 as terminals. A21 gives only 1,3,4,5 as terminals. So not.

Thus insertion for 14 and 21 necessary.

Now for the "17 15 16" after 23: maybe some production already includes that sequence partially, e.g., A23 is defined with A24 A23 (where A24 produce 18, 19, 20). Not helpful. A16 may produce "17 15 16"? Actually 17 is terminal in many productions like "A68 -> 17". 15 maybe produced by other nonterminals like A15 -> 1 etc. But not together.

Thus insertion needed.

Now after A23 we need 19 20 18 1; but these are produced by A23's recursion as described.

Now after 22 we need '17'. We need to insert that.

Thus insertion for middle segment total is 6.

Now suffix segment insertion count we determined 3.

Thus minimal total insertions: 3 (initial) + 6 (middle) + 3 (suffix) = 12.

But perhaps we could reduce middle insertions by using both A14 and A37. For instance, we could have A14 produce "23 A23 22" with two insertions for "14 21". Then after that we could have A20 produce suffix plus we could have an extra nonterminal after 22 that yields "17" without insertion. For example, if after A14 we have A20 (we already have). So A14's production ends with terminal 22; then we apply A20 (which starts with 9). There's no need for 17 after 22? Actually target wants "17" after 22 before the suffix starting with 9. So we need a 17 terminal before the suffix.

So we could modify A14's production to include "22 17" (by insertion of 17 after 22). That's one insertion. So middle insertion count would be:

- insert "17 15 16" after 23 (3)
- insert "14 21" after A23 (2)
- insert "17" after 22 (1)

Total 6 exactly as before. So 6.

Is there any way to combine some of those with other productions so that we can share insertions? For example, maybe we could add "17 15 16 14 21 22 17" as a block in a single insertion? Wait we cannot replace existing terminals, but we could insert a sequence of terminals in a production. If we insert after 23 the entire sequence "17 15 16 A23 14 21 22 17"? Actually we cannot insert nonterminal A23. But we can insert terminals. The production originally is "23 A23 22". We could insert terminals before A23, after A23, and after 22. That's three insertion points. At each insertion point we can insert multiple terminals.

Specifically we could:

- After "23": insert "17 15 16". (3 terminals)
- After A23: insert "14 21". (2 terminals)
- After "22": insert "17". (1 terminal)

Thus the total inserted terminals = 3+2+1 = 6. So minimal for that part is 6 unless we find another nonterminal with a production that already includes some of them.

But there is rule 702 (A15 -> 23 17 15 16 A23 17) which includes core segment "23 17 15 16 ... 17". It lacks 14 21 22. Could we use A15 to generate the segment followed by A22 etc? Let's examine.

In A14 -> A19 A15 (rule 235). So A14 could produce A19 A15. If we set A19 to produce "A23 14 21 22"? Maybe A19 -> A14 A20 ... hmm.

Alternative path: Use A15 production to produce "23 17 15 16 A23 17". Then we can insert "14 21 22" after A23 (maybe at that point). That would give "23 17 15 16 A23 14 21 22 17". That yields exactly: 23 17 15 16, then A23 produces 19 20 18 1, then we insert 14 21 22, then final 17. Wait the order: "A15" gives "23 17 15 16 A23 17". Insert after A23 "14 21 22". That would place 14 21 22 before the trailing 17, which is correct. So overall after insertion we have: 23 17 15 16 A23 14 21 22 17. That's exactly the needed pattern! So using rule A15 is better because it already gives 23 17 15 16 and the final 17.

Thus we can use A15 instead of A14. Let's examine where we can get A15 in the derivation after A9 block.

Remember after A9 block we go to A19. A19 can be expanded as A14 A20 (rule 260) or as ... maybe there is a production that yields A15 as a child. Let's examine A19 expansions:

- A19 -> 1
- A19 -> A7 A5
- A19 -> A8 A19
- A19 -> A14 A20
- A19 -> A19 A18

None give A15 directly. But we could use A7 A5 route: A7 can lead to A15 maybe. Let's examine A7-> A14 A11 (rule 198); A11 can expand to A20 A16; not A15.

Alternatively A7-> A7 A6 etc. Not helpful.

But perhaps we can use A19 -> A8 A19 (rule 259). A8 expansions could produce A15? Let's check A8 rules: 200: A8->1, 201: A8-> A7 A9, 202: A8-> A8 A8, 203: A8-> A14 A10, 204: A8-> A19 A17. No A15.

Therefore A15 may not be reachable via A19 directly.

But maybe we could use A5 -> A9 A19 (rule 188). In A9 we did. After that, we have A19 (RHS). Perhaps we could restructure the derivation to use A9 then A14-like path. However we could use A9's product plus then tail: A19 can be something else like A7 A5 etc. Let's examine A19 -> A7 A5 (258). Then A5 can eventually produce A13 A20 (rule 189). A13 can produce A6 A13 etc. But perhaps we can produce A15 via expansions from A5? Let's see A5's productions: rule 185 (A5 -> 1), rule 186 (A5 -> A5 A18), rule 187 (A5 -> A6 A5), rule 188 (A5 -> A9 A19), rule 189 (A5 -> A13 A20). So A5 does not directly produce A15. So not.

Thus we might not be able to incorporate A15 directly unless we go deeper.

Alternatively, we could use A19 -> A14 A20, where A14 -> A19 A15 (via rule 235), which could bring A15 into the derivation after A14. Let's examine:

A14 -> A19 A15 (rule 235).

Thus if we choose A19 -> A14 A20 (260) and then expand the A14 as A19 A15 (235). So the derivation sequence becomes:

A19 -> A14 A20
 A14 -> A19 A15
 So we have A19 A15 A20.

Then we need to generate A19 (first) and A15 and A20. This introduces A19 again, which can expand to ... maybe we can have A19 -> 1 (or other), but we need the first A19 to generate the "23 ..." segment? Let's think:

Goal: after the block from A9, we need to produce "23 17 15 16 ...", which is captured by A15's production. So we could have the sequence:

A19 -> A14 A20
A14 -> A19 A15

Thus the overall yields: A19 (first) then A15 then A20.

If we let the first A19 be something that generates nothing (maybe epsilon via A19->1?). Actually A19->1 yields terminal 1; that's not needed in this segment (target after 28 27 is "23 ..."). There's a "1" before 14? Wait target after the first segment includes "23". No "1". So cannot have A19 produce 1 there.

However we need to produce nothing before A15; we could modify A14's production to produce A15 directly, but using rule 235 includes A19 A15. If we can make A19 produce epsilon (empty) via insertions? No, can't delete symbols. But we can insert terminals; cannot remove the 1. So not possible.

Alternatively, we might consider using A19 -> A14 A20 and use A14's other production (236) to produce "23 A23 22" and then rely on A20 for rest. This path (A14 -> 23 A23 22) plus A20 (which includes suffix). That requires inserting 14 21 after A23 and 17 after 22, etc., as previously estimated (6? Actually earlier considered 6 insertions for middle plus suffix, but we might be able to handle suffix via A20 (so after 22 we need a 17 before A20; we can insert that, then use A20 for suffix). So we still need 6 insertions for the middle segment.

But we can reduce insertions perhaps by using A15 for the 23 17 15 16 etc part, and combine with A20 for suffix. Let's examine route: Have A19 -> A15 A20 (choose some rule). Is there a production allowing A19 to be A15 A20? Not directly. However we could use A19 -> A14 A20 (260) and then set A14 -> A15 via a different production? Not; A14 cannot expand to A15 directly. However A14 -> A19 A15 includes A15 but also A19 again.

Could we apply A14 -> A19 A15, then have that first A19 produce directly A15 A20? That's recursive and complex.

Potentially we could derive the pattern as: A19 -> A14 A20; A14 -> A19 A15; then the inner A19 could be expanded to 1 (terminal), or to something else. If we expand inner A19 to A14 A20 again, leads to recursion. Not helpful.

But we could possibly set A14 to use rule 236 (23 A23 22) and we keep A20 after that. This requires insertions.

Thus seems the 6 insertion count for middle segment is minimal. But we might be able to reduce the 2 insertions for 14 21 by using some existing production that already yields "14 21". Let's check if there is any production where terminals "14 21" appear consecutively. Search list: At rule 474 we have "... 14 21 22" (that's good). Also A41 perhaps? Not.

Thus we could combine insertion to put A23 before "14 21 22". Actually rule 474 already provides "23 A23 14 21 22". So we could use A37 with rule 474, but we need 23 17 15 16 before 23. However we could insert "17 15 16" before the 23 generated by A37? Wait rule 474 has "23 A23 14 21 22". That's after the 23. The required pattern is "23 17 15 16 ..." The 17 15 16 should be after the 23 and before A23. So we could use rule 474 and insert "17 15 16" after the 23, before A23. That would be 3 insertions. So using A37 reduces the insertion count from 6 to 4? Wait earlier we said using A14 inserted 6 terminals; using A37 we need to insert: after 23, insert "17 15 16" (3). Then after A23 (which yields 19 20 18 1) we need to insert "??" Since rule 474 already includes 14 21 before 22, so we don't need to insert those. Then need to insert final 17 after 22 (1). So total insertions = 3 + 1 = 4. That's better.

Thus we must generate A37 in the derivation.

Now how can we get A37? The nonterminal A37 appears as a distinct symbol. Could we have A19 produce A37 A20, or something? Let's check A19 alternatives: they are: 1; A7 A5; A8 A19; A14 A20; A19 A18. None produce A37. So we need to generate A37 via other path.

Perhaps we could generate A37 using A5 or other expansions. Maybe we can substitute A14 with A37 by using A14 -> A14 A12 or etc. Not.

Maybe we can generate A37 from A4 -> A5? No.

But perhaps we can "insert" the production A14 -> A37 (as a terminal insertion) but we cannot change LHS and RHS nonterminals, only insert terminals. So cannot.

Thus we need A37 to appear as a child of some nonterminal that we can "navigate to". Could be A7 -> A14 A11 (not). A8 -> A14 A10 (rule 203). A14 appears often but doesn't transform to A37.

Perhaps we could reach A37 through A18? A18 expansions include A15 A20 etc., maybe produce A37 later via A20? Not.

Alternatively, we could have A5 -> A13 A20 (rule 189). A13 may expand to something that includes A37? Let's examine A13 productions: A13->1 (226), A13-> A5 A15, A13-> A6 A13, A13-> A9 A14, A13-> A13 A12. None produce A37.

A6 productions: none produce A37.

Thus not.

What about other path: A2 -> A61 A3, A3 -> A27, A27 -> A41, A41 -> A38 A43, A38 -> A41 A38 etc. Somewhere A27 could produce A41 which could produce A38 which could generate A37? Actually A38's productions include "A38 -> A41 A38". but we could have A38 -> A41 A38, which leads to loops but not produce A37.

But A38 is also used in A41 -> A38 A43. So A41 can produce A38 then A43. A38 can produce A41 A38 (recursion) or A39 A36, A40 A37, etc. So A38 can produce A40 A37 (rule 483). So that path can generate A37 inside later.

Thus we can try to generate A37 by using A41 -> A38 A43, and within A38 choose rule 483: A38 -> A40 A37. This will generate A40 (nonterminal) and A37. Then A40 may generate something (maybe produce the 14 21 22?) Actually we may not need A40 if we rely on A37 for big segment. However we also have A43 after A38 (if we still use A41 -> A38 A43). The order would be: A40 (some expansions) then A37 then A43 (suffix). But we need the segment after A9 block: "23 17 15 16 ... 14 21 22 17 9 7 10 11 ...". A37's segment (with insertions) will produce the part from 23 to final 17 (including needed), then after that we need A43 (or A20) to produce suffix. So indeed using A41 -> A38 A43 with A38 -> A40 A37 may give A40 then A37 then A43. But we need A40 to not disrupt sequence. We could insert terminals in A40 productions or prune them? But we cannot delete; we could insert terminals. However maybe we can have A40 generate empty (or just 1). Let's examine A40 productions: rule_490: A40 -> 1 ; rule_491: A40 -> A38 A33 ; rule_492: A40 -> A39 A32 ; rule_493: A40 -> A40 A31 ; rule_494: A40 -> A41 A40.

So base production for A40 is terminal 1. So if we pick A40 -> 1, then A40 will output terminal "1". That would introduce an extra "1" before the A37 segment, which target does not have (there is no extra 1 after the previous 27? Wait let's see the target after ... 27 (pos14). The next token pos15 is 23, not 1. So an extra "1" would be extra. We cannot delete it; we could insert to compensate? It would mess up order. So perhaps we avoid using A40 by making A38 use a different production that avoids generating A40? Let's see: other A38 productions: A38 -> 1 ; A38 -> A38 A35 ; A38 -> A39 A36 ; A38 -> A40 A37 ; A38 -> A41 A38. The production A38 -> 1 would generate a 1 before A37 (if we instead do A38 -> A41 A38 then eventually get A38 -> ... not). But maybe we can choose a path that does not involve A40. Look at A38 -> A39 A36: this yields A39 then A36 before A37 (not needed). Could we generate A37 directly from A38 without other terminals? Not via given productions. The only path to get A37 from A38 is via rule 483 (A38->A40 A37). Thus we have an unavoidable extra "1" from A40 unless we choose A40 to generate something that may be eliminated via insertions? But we cannot delete 1, but maybe we can "absorb" it into target if we insert a different terminal before it, maybe we can treat the extra 1 as acceptable if we also insert something else later to shift? Not possible since insertion only adds terminals, not remove.

Thus using the A41/A38 route injects an unwanted 1 at start of A37 segment, which would not match target.

We could also generate A37 via some other nonterminal path not requiring A40. Let's see other occurrences of A37: It appears on RHS of many productions: A15 -> A16 A13 (239) etc. Or A17 -> etc. But A37 itself appears as a terminal nonterminal that could be directly invoked if some other rule expands to A37. Let's search for productions that contain A37 as RHS directly: yes e.g., rule_474 is A37 itself, not RHS. Let's check where A37 appears as RHS:

- rule_477: A37 -> A31 A37 (RHS includes A37)
- rule_468? Not.
- rule_483 (A38 -> A40 A37)
- rule_476: A37 -> A30 A36
- rule_475? Not.
- rule_479: A37 -> A42 A38 (RHS contains A42 A38)
- rule_473: A37 -> 1
- rule_474: not.

Also rule 239: A15 -> A16 A13 - not.

Thus A37 appears on RHS in A38 -> A40 A37; also maybe in other nonterminals like A30? Not: A30 has production leading to A42 A39 (but includes A42). A31 -> A30 A32 etc. So the only direct RHS occurrence is via A38.

Thus we seem stuck; we can't get to A37 without going through A38 and thus generating an extra 1 via A40->1. However, maybe we can use A40 not as terminal 1 but as something that later leads to something else that we could insert something to handle? But we will still have a terminal from A40 that is maybe not 1: Wait rule 491: A40 -> A38 A33 (so A40 can expand to A38 A33). That is recursive; again not terminal. But A40's base production is terminal 1 (rule 490). So any derivation will include at least one 1 from A40 (unless we use infinite recursion, but must terminate). So we can't avoid that 1. That would cause an extra 1 before A37 segment, which is not in target.

But maybe we can arrange that the extra 1 corresponds to the target's "1" that appears after position 22? Wait we have a '1' in target after 18 (pos 22). Let's line up target positions to see where the extra 1 could be placed.

Target positions after initial block (positions 1-14: 1 1 30 27 13 29 24 26 25 1 12 31 28 27) then position 15 is 23 etc.

Note we have a 1 after 27? Actually there is a 1 earlier at position 10 (from A25). Then after 28 27 we have 23... There's no extra 1 before 23. So the extra 1 from A40 would be placed before the 23 segment, which would not match target. So we cannot go that route.

Thus we must avoid generating A37 at all, using A14 with insertion.

Thus minimal insertion solution appears to be using A14 -> 23 A23 22 with insertions: after 23 insert 17 15 16; after A23 insert 14 21; after 22 insert 17. Total 6.

Now overall total insertions = 3 (A9 modifications) + 6 (A14 modifications) + 3 (A20 modifications) = 12.

But we may be able to reduce by also generating 14 21 directly elsewhere, perhaps within A20's suffix or earlier? However that segment is after 22 before 17. Could we produce 14 21 as part of suffix via A20? No, A20 generates 9 7 10 ... not 14 or 21. So not.

Could we produce 14 21 via A18 insertion? Not.

Thus 12 may be minimal given constraints.

Now we must produce final updated production rules which reflect added terminals (inserted). We must list only changed rules with insertions.

Thus we need to modify the following rules:

- rule_210: Insert '27' after 30, then insert '12' after A25, insert '31' after 12 (or after A25 can be "A25 12 31 28 27"). So final RHS: "30 27 13 29 A25 12 31 28 27". Wait also we inserted 27 after 30 but then we need to keep original order of 13 29 and others. So final: "30 27 13 29 A25 12 31 28 27". That's three terminals inserted.

But note we removed original trailing "27"? Actually original rule: "A9 -> 30 13 29 A25 28 27". We'll have "30 27 13 29 A25 12 31 28 27". Good.

Thus rule_210 updated.

Now rule_236: "A14 -> 23 A23 22" currently. We need to insert "17 15 16" after 23, insert "14 21" after A23, and insert "17" after 22. So RHS becomes: "23 17 15 16 A23 14 21 22 17". Wait we also need to keep original 22 terminal before the final 17. So after 22 we insert 17. So final: "23 17 15 16 A23 14 21 22 17". That's 6 inserted terminals: 17,15,16 after 23; 14,21 after A23; 17 after 22.

Thus rule_236 updated.

Now rule_263: "A20 -> 9 7 10 A21 6 7". Insert "11" after "10". Insert "8 2" after "6". So final RHS: "9 7 10 11 A21 6 8 2 7". Wait we insert "11" after 10; and "8 2" after 6. So final: "9 7 10 11 A21 6 8 2 7". That's 3 inserted terminals: 11,8,2.

Thus rule_263 updated.

Are there any other modifications needed? Perhaps we also need to ensure that the insertion of "31" is a terminal 31. That's fine.

Now we must verify that after modifications the grammar can indeed derive target.

Let's simulate derivation after modifications.

We'll present derivation skeleton:

A1 -> A2 -> A61 A3

A61 -> A63 A62 (choose rule 738). Both produce terminal 1 each: A63->1, A62->1 => "1 1".

Now A3 -> A4 (choose rule 38). A4 -> A5 (choose first rule among many A4 -> A5). A5 -> A9 A19 (rule 188). A9 now expanded with modifications: rule_210: 30 27 13 29 A25 12 31 28 27

Now A25 expansions produce: Choose A25 -> A26 A25 and recursively produce 24,26,25 and finally 1 via A25->1.

Thus A5's output: 30 27 13 29 24 26 25 1 12 31 28 27.

Thus after A5, we have the sequence up to token 14 correct.

Now after A5, we need A19. A19 is remaining after A5 -> A9 A19.

A19 -> A14 A20 (rule 260). A14 with modification: 23 17 15 16 A23 14 21 22 17.

Now A23 expands into sequence "19 20 18 1" (choose A23 -> A24 A23 with A24->19; then again A24->20; then again A24->18; then finally A23->1). So A23 yields 19 20 18 1.

Thus A14 yields: 23 17 15 16 19 20 18 1 14 21 22 17

Now A20 with modification: "9 7 10 11 A21 6 8 2 7". A21 can recursively produce a sequence: choose A21 -> A22 A21 with A22 ->5 (first), then A21 -> A22 A21 with A22->3, then A21 -> A22 A21 with A22->4, then A21 -> 1. Thus A21 yields 5 3 4 1.

Thus A20 yields: 9 7 10 11 5 3 4 1 6 8 2 7

Combine all sequences:

1 1 (from A61) + (30 27 13 29 24 26 25 1 12 31 28 27) + (23 17 15 16 19 20 18 1 14 21 22 17) + (9 7 10 11 5 3 4 1 6 8 2 7)

Let's verify matches target exactly:

- First 2 tokens: 1 1 - match.
- Next: 30 (match), 27 (match), 13 (match), 29 (match), 24 (match), 26 (match), 25 (match), 1 (match), 12 (match), 31 (match), 28 (match), 27 (match) – yes.

- Next: 23 (match), 17 (match), 15 (match), 16 (match), 19 (match), 20 (match), 18 (match), 1 (match), 14 (match), 21 (match), 22 (match), 17 (match) – yes.

- Next: 9 (match),7 (match),10 (match),11 (match),5 (match),3 (match),4 (match),1 (match),6 (match),8 (match),2 (match),7 (match) – yes.

Thus entire target covered exactly.

Now we need to output only the updated production rules with their rules numbers.

Thus include:

- rule_210 updated: "rule_210: A9 -> 30 27 13 29 A25 12 31 28 27".

But note need to use the format "rule_<n>: ..." exactly.

- rule_236 updated: "rule_236: A14 -> 23 17 15 16 A23 14 21 22 17". (Check the order: Insert 17 15 16 after 23, then A23, then 14 21, then 22, then 17). So the RHS is "23 17 15 16 A23 14 21 22 17". yes.

- rule_263 updated: "rule_263: A20 -> 9 7 10 11 A21 6 8 2 7". (Check order: 9 7 10 11 A21 6 8 2 7).

Now produce final answer with these updated rules. Ensure no extra text after.

Thus final output: three lines, each with rule number and updated RHS.

Let's make sure we don't miss any numbers: The rule numbers for modifications are 210, 236, 263.

Check that rule_236 originally "A14 -> 23 A23 22". So after insertions, we have "23 17 15 16 A23 14 21 22 17". Good. Did we insert the numbers "17 15 16"? Yes inserted after 23, before A23.

But note we also need to keep original 22 before final 17 (which we have). So final is "23 17 15 16 A23 14 21 22 17". Good.

Now for rule_210: originally "A9 -> 30 13 29 A25 28 27". Insert 27 after 30, 12 and 31 after A25. So final: "30 27 13 29 A25 12 31 28 27". Works.

Now rule_263: originally "A20 -> 9 7 10 A21 6 7". Insert 11 after 10 and 8 2 after 6, final "9 7 10 11 A21 6 8 2 7".

Now, do we need to ensure that terminal 31 is indeed allowed (1..31)? Yes 31 is allowed.

Everything good.

Potential issue: We inserted terminal 27 after 30; but we also still have the original 27 at the end. So total two 27 in that block: "30 27 13 29 ... 28 27". Target has at positions 4 = 27 and later at 14 = 27. So two 27's present. Good.

Now after this block, there is also a 27 in target at position 14 (the final 27 of block). So matches.

Thus we are done.

Check if any other point of missing terminals might need insertion: For example, we used 14 21. Those are produced after A23 via insertion. Terminal 14 appears also later in the target, after the 1 at pos22: indeed target includes "14 21 22". Good. So we inserted them correctly.

Now we need to confirm that we haven't double used 14 elsewhere (maybe used already in other expansions). 14 is also a terminal elsewhere (A44?). But fine.

Now final answer.Updated production rules:
rule_210: A9 -> 30 27 13 29 A25 12 31 28 27
rule_236: A14 -> 23 17 15 16 A23 14 21 22 17
rule_263: A20 -> 9 7 10 11 A21 6 8 2 7