We need to understand the problem: given a CFG with many production rules (probably artificially constructed), the target sequence is a list of terminals 1..34 (which are integers). The grammar initially had many terminal symbols removed. The task is to insert terminal symbols into production rules (only insertion, no deletion or reordering; new production rules can't be created) to make the target sequence derivable from the start symbol A1. We need to minimize the total number of inserted terminal symbols across all rules.

The tricky part: The grammar has a huge amount of redundancy: many rules for A2 -> A78 A3 repeated many times (almost 43 times approx). Some are missing terminals? Possibly the target sequence originally was produced by some particular choice of terminals in those rules, but they got removed. So we need to restore them by inserting terminal symbols where needed.

We can insert any terminal symbol in any rule at any location. We cannot reorder or delete symbols. So we can adjust each RHS by adding terminals.

Goal: derivation from A1 to target sequence: w = "1 1 34 31 14 33 29 28 27 26 1 13 32 ..."

Let's rewrite target sequence clearly:

Indices (1-indexed):
1: 1
2: 1
3: 34
4: 31
5: 14
6: 33
7: 29
8: 28
9: 27
10: 26
11: 1
12: 13
13: 32
14: 30
15: 31
16: 25
17: 21
18: 16
19: 24
20: 19
21: 17
22: 18
23: 20
24: 1
25: 15
26: 23
27: 22
28: 21
29: 12
30: 8
31: 9
32: 10
33: 3
34: 5
35: 4
36: 2
37: 1
38: 7
39: 11
40: 6
41: 8
42: 1

Sequence length: 42 terminals.

We need the grammar to produce this exact sequence. There may be multiple derivations; we can insert terminals in rules to match.

Given the huge number of rules, many potential productions generate terminals already: like in many rules we have explicit terminals, e.g., rule_326: A5 -> 1; rule_331: A6 -> 1; etc. Also for composite rules like A49 -> 34 14 A25 32 30 (contains terminals 34,14,...). And many rules like A49 -> 34 14 33 A25 13 31 etc. Also A49 -> 34 A25 30.

Various rules also have sequences like A51 -> A50 A47, etc.

Our target appears to have a pattern: first two 1's: this could be derived from A5 (1) then maybe A5 again, or other nonterminals that produce 1.

But consider the start: A1 -> A2 (rule_1). Then A2 -> A78 A3 (rules_2 to 45 etc). So A2 is (A78 A3). So A1 -> A2 -> A78 A3.

Now, A78 likely leads to productions of certain terminals. Let's look at the rules for A78, A79, etc: We have rule_1450: A78 -> A80 A79. Then rule_1451: A78 -> A83 A81. rule_1452: A78 -> A86 A84. rule_1453: A78 -> A89 A87. So A78 expands to one of those pairs.

Goal: produce the target sequence likely using those expansions.

Firstly, there are 42 terminals. Likely the grammar is designed to produce a long sequence using many nonterminals that produce sequences of numbers 1..34.

Existing productions that directly contain the target terminals include some long sequences: such as rule_351: A9 -> 34 31 A25 13 32 (includes 34,31,... something). Rule_408: A20 -> 12 A21 6. Rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 (multiple). Also A43 -> 12 8 10 A21 11 8 (staged). There is also A61 production with lots of A62 etc but no explicit terminals. Some rules produce combos like A77 -> 12 8 9 10 A21 (some subset). And A77 -> ... Another: A71 -> 25 A23 22 or 25 A23 23.

Thus many sub-sequences in target may be derived from these.

Given the complexity, we need to decide how to insert minimal terminals. Usually the minimal approach is to find an existing derivation that yields the target except missing some terminals (maybe those removed). The repetition of many identical productions for A2 -> A78 A3 suggests that perhaps some terminals originally existed in these rules, and all but one instance may have had the same removed terminals. Since they are all exactly the same currently (just A78 A3), maybe they need to have some terminal inserted to generate the needed sequence.

Similarly, A4 -> A5 repeated many times. Actually from rule_70 onward, many duplicate rules A4 -> A5 are repeated many times. Then A4 -> A18 also repeated many times. Then A4 -> A19 repeated many times. Then A4 -> A20 repeated many times. So A4 may have multiple possible expansions. This heavily duplicates. So to produce an arbitrary long sequence, we need to use those expansions in a particular order.

Given the start, A1 -> A2. Then A2 -> A78 A3 (lowest). So we need to produce the entire target using A78 and A3 expansions.

Let's examine A3 productions: rule_66: A3 -> A4; rule_67: A3 -> A27; rule_68: A3 -> A44; rule_69: A3 -> A61. So A3 can become A4 (which is numerous expansions), A27 (lots of expansions), A44 (again many repeats), and A61 (which expands heavily to A62 and more). So there are many possibilities.

Thus the beginning A78 A3 must produce 42 terminals.

We need to find a path with minimal inserts.

Given the complexity, the best approach is to treat the given grammar as akin to a universal or nearly universal generator: each nonterminal may produce any possible sequence of terminals by recursively decomposing into other nonterminals and some terminal strings. The target includes numbers up to 34. The existing productions include many of these numbers as terminals automatically: e.g., 34 occurs directly in A90? Actually rule_324: A4 -> A5 (nonterminal). Then rule_326: A5 -> 1. So A5 -> 1 is terminal 1. There are other direct terminal productions for many numbers, like rule_326 only gives 1. Where do terminals like 34 appear? A5 doesn't contain 34. Some rules like A9 -> 34 31 A25 13 32 produce 34 and 31 and 13 and 32. Also A49 -> 34 14 A25 32 30 etc. Also A49 -> 34 14 33 A25 13 31 includes 33 as well.

Thus all target numbers could be derived from these large sequences.

Given the target sequence includes [1,1,34,31,14,33,29,...], we can try to map subsegments to specific productions:

Start: 1,1: could be two A5 -> 1 sequential expansions. Or maybe A2 (or A78->something) produce 1??

A2 -> A78 A3: maybe A78 produces 1 then A3 produces the rest: "1 34 31 14 33 ...", etc. Or A78 may produce the first "1 1"? But unlikely.

Let's examine A78 productions:

- A78 -> A80 A79 (rule_1450)
- A78 -> A83 A81 (rule_1451)
- A78 -> A86 A84 (rule_1452)
- A78 -> A89 A87 (rule_1453)

Thus A78 is a pair of sub-nonterminals; each of those may generate sequences.

Let's examine each left sub-nonterminal:

A80: rule_1479-1482: A80 -> 1 (repeated four rules). So A80 directly produces terminal 1.

Thus A78 -> A80 A79 yields first 1 then whatever A79 produces.

A79: many rules rule_1454-1479 show A79 -> 1 (back to back many duplicates). The repeated rules for A79 all have RHS "1". For each rule, A79 directly produces terminal 1. No other expansions for A79.

Thus A78 -> A80 A79 yields "1 1". Great: covers first two 1's in target.

Alternatively, A78 can be other combos.

Now after those, we need to produce the rest (starting from 34). So A3 must produce the remaining sequence after the two ones.

Thus likely the correct derivation is: A1 -> A2 -> A78 A3, where A78 -> A80 A79 yields "1 1" (since each produce 1). Then A3 must generate the remaining 40 terminals.

Now we need to find a derivation for A3 to produce the remaining sequence: 34 31 14 33 29 28 27 26 1 13 32 30 31 25 21 16 24 19 17 18 20 1 15 23 22 21 12 8 9 10 3 5 4 2 1 7 11 6 8 1

Let's denote suffix S = [34,31,14,33,29,28,27,26,1,13,32,30,31,25,21,16,24,19,17,18,20,1,15,23,22,21,12,8,9,10,3,5,4,2,1,7,11,6,8,1] (40 tokens). We need to generate S from A3.

Recall A3 -> A4, A27, A44, A61. So we can consider which leads to S.

Given that A4 has many expansions: either to A5 (which yields 1), to A18 (some complex sequences), to A19, to A20. Maybe we can produce S using concatenations of those expansions.

Another path: A27 is heavily ambiguous; probably can generate a lot of numbers like 29,28,... oh wait A27's productions only go to A28 or A41; but does not directly produce terminals. So likely A27 leads to deeper nonterminal expansions.

A44 leads to many expansions to A45 (which maybe yields more). A61 expands to many A62 etc which may produce sequences.

But we need to find a minimal insertion scenario: some rules already have needed terminal sequences (like A9 -> 34 31 A25 13 32 contains 34,31,13,32). Our target after 33 has sublist 29,28,27,26,1,13,32 ... So we might need to embed 34 31 from A9 and maybe other numbers from other rules.

But maybe there is a simpler path: Maybe A3 -> A4 and then A4 -> A5 (→1) repeated many times to produce the rest? But we need numbers beyond 1.

A2 has many expansions all identical: A2 -> A78 A3. So A2 always expects just one A78 then A3. So the rest of the sequence cannot be produced by A2 after A78.

Thus A3 must produce the entire remaining S.

Now consider A3 -> A61. A61 has many productions (A62, A75, A76). But those productions have many recursion expansions but have few explicit terminals: A62 -> 1; likewise A63 -> 1; many others produce 1 and recursive expansions: these nonterminals likely generate sequences purely of 1's (if no external terminals). However some also have productions with larger numeric terminals: e.g., A66 includes a RHS with "34 14 33 A25 13 32". A66 is in the chain A62 -> A63? Let's see.

A62 -> A63 A62 (rule_1368). Actually A62 can expand via rule_1368 to A63 A62. A62 also can produce `A66 A76` (rule_1369) and `A70 A77` (rule_1370) and also `A62 A75` (rule_1367) and `1` (rule_1366). So A62 can directly produce 1, or recursively produce large combos.

Also A62 -> A66 A76 may be helpful: A66 has a production with many terminals (including 34 14 33 etc). So A61 could go to A62 and then to A66 which yields some target subsequence.

Thus we need to check many productions for numbers.

Let's trace a plausible derivation for S using A3 -> A61 -> A62 -> A66 A76; then A66 yields the start of S: maybe includes 34,14,33, ... but we need 34,31,14,33... Hmm A66's production: rule_1387: A66 -> 34 14 33 A25 13 32

Sequence: 34,14,33,(A25),13,32.

Target prefix for S: 34,31,14,33,... So we need 31 after 34, before 14. In A66 we have no 31. However there's rule_1386: A66 -> A63 A66 (recursive), A66 -> A66 A65, etc. But not 31.

Alternatively, A9 -> 34 31 A25 13 32 (rule_351) gives 34,31, (A25),13,32 exactly matches S's first three numbers: 34,31,14? Wait 31 is second; but we need sequence 34,31,14,... Not 34,31,13... Actually A9's RHS is 34 31 A25 13 32. This yields 34,31,(expansion of A25),13,32. After expansions of A25, we need 14? We have 14 after 31 in target. But A9 has no 14 before the A25. The 13 appears after A25 (like after some unknown expansion). So maybe we need to insert 14 after A9->... e.g., modify rule_351 to include 14 terminal between 31 and A25? Potentially.

But we can also insert terminals in any rule. So maybe we can adapt existing rules by adding missing terminals (the removed ones). So the goal is to minimize number of insertions required to allow the target to be generated. Because we can insert terminals anywhere, we likely need to insert some where they were removed: maybe in A2 -> A78 A3 (or in A4 -> A5) many times.

But maybe we only need to insert a few at places where the target's numbers are not already present in any productions. Since all terminals appear somewhere else: all 1..34 appear among rules. Let's check each number: 1 appears many. 2 appears as A22 -> 2 (rule_412). 3 appears as A22 -> 3 (rule_413). 4 appears as A22 -> 4 (rule_414). 5 appears as A22 -> 5 (rule_415). 6 appears as rule_378? Actually A11->??? but A6->1 so 6 not as terminal? Actually we see A6->1, not 6. We need terminal 6 (the integer 6) appears in A45? No, let's search: Terminal numbers appear in rules: many rule names contain numbers but not terminals. Terminals are literal numbers. For example, rule_336: A7 -> 1. So many terminals are 1 only. But other numbers appear: rule_378 A15 -> 1 (again). Many productions use numbers >1: Eg A21->1, but also A22->2,3,4,5. A24->17,18,... So numbers up to maybe 34 appear in some productions. Let's list occurrences:

- 1 appears numerous places.
- 2 appears in rule_412 (A22 -> 2): So we can produce a 2 terminal from A22.
- 3 appears (rule_413): A22 -> 3.
- 4 appears (rule_414): A22 -> 4.
- 5 appears (rule_415): A22 -> 5.
- 6 appears? Maybe as a terminal number in some productions like A20-> A20 -> A20 -> ... There's rule_1510: A61 -> A62. Not there. Let's search for "6" not as a rule number but as RHS literal: For instance, rule_306: A4 -> A20 (no 6). But look at A21: rules 411: A21 -> A22 A21; left side uses A22 which might produce 2,3,4,5; but no 6.

Search "6 " in productions: rule_1510 something? Actually there are many occurrences like "6" in A20 productions: rule_406: A20 -> 12 9 10 A21 8, has terminal 8 but not 6. rule_408: A20 -> 12 A21 6 (includes 6). So terminal 6 appears here. Also rule_438 maybe? Not needed.

Now check 7 appears: e.g., A22 -> ??? Actually 7 appears in rule_420? A24 -> 20 (no). 7 appears as terminal in A21: A21 -> ??? No. but rule_418: A24 -> 17 (terminal 17). So maybe 7 appears as terminal in A45? Another: rule_1515? I'd need to scan for 7 literal:

- rule_308: A4 -> A20 (no). We need direct presence: For example, rule_161: A1 maybe not. Let's quickly search manually: we have "7" appears in rule_408: "12 A21 6". That's 6 not 7. rule_403: A20 -> 1. (no). rule_404: A20 -> A10 A19. Not. rule_405: A20 -> A11 A5. Not. rule_406: A20 -> 12 9 10 A21 8. Contains 12,9,10,8. Not 7. rule_408 again. rule_409: A20 -> A12 A20. no. rule_410: A21 -> 1. not. rule_411: A21 -> A22 A21: A22 could produce 2-5; not 7. Then later at the end, rule_1445: A43 -> 12 8 9 10 A21 (no 7). rule_1445 does not contain 7. rule_1444 maybe? Actually rule_1445 includes "12 8 9 10 A21". There's also rule_1442: maybe includes other numbers.

Thus terminals 7 and others appear in some productions at the very end? Also A14 -> 1 (no). But A71 -> 25 A23 22 and 25 A23 23. So terminals 22,23 etc appear.

We need to produce all numbers 1..34 from the grammar, but it seems possible.

Now the problem: we need to minimally insert missing terminals to generate the exact target string. Many terminals already appear in productions but may not be arranged in correct order positions. Insertions will fill in missing numbers between known tokens.

Thus, we need to look for a sequence of production expansions that yields S if we insert some terminals.

Given the language is extremely powerful (maybe Turing-complete), possibly any finite string can be derived via appropriate insertions. Minimizing insertions is reminiscent of editing the grammar to fill gaps where terminals are missing.

Given the huge redundancy, perhaps the optimal solution is to insert all the target terminals into the one rule that can produce the whole string: For example, rule_1: A1 -> A2. Could we insert the entire target in rule_2: A2 -> A78 A3? But we can't reorder symbols, only insert. To produce the target we can insert after A78 and A3? However we need the order. Currently A1 -> A2. Perhaps we could modify rule_2 (or any rule for A2) by inserting the whole target sequence into the RHS, but we must preserve the ordering of existing symbols (A78 A3). If we insert the target before or after them, we will get extra nonterminals that produce other symbols; however, we need to generate the target only; we could delete the nonterminal expansions? But deletion is prohibited. So we cannot remove the A78 A3 that would further derive extra symbols unless they derive empty (epsilon). But they cannot derive empty because no epsilon productions exist (there's no rule producing empty). So they must produce something; thus we cannot have them remain unless they produce correct terminals as part of target.

Thus we need to fill directions for all nonterminals used (A78 and A3) to generate segments that together with the target fill exactly the whole sequence.

Thus we need to find expansions for A78 and A3 (and other required nonterms) that produce the target without extra terminals beyond those inserted.

Goal: produce "1 1" from A78, maybe use A78 -> A80 A79, each -> 1. This yields exactly "1 1". So no insertion for A78 needed.

Now A3 must produce the rest 40 tokens.

Thus we need to fill expansions for A3 (choose production A3 -> A4 or A27 or A44 or A61). For each we may need to insert some terminals.

Given the complexity, we need to search for a production path that can generate the rest with minimal additions.

It may be that the original grammar had more explicit terminals in many of the repeated rules for A2 -> A78 A3 etc, but they were all stripped. So maybe the original version had a specific fixed sequence of terminal symbols interspersed in those identical rules. For instance, there are 43 identical rules A2 -> A78 A3. Perhaps originally each of those rules had different digits inserted (like hooking each terminal in the target). Similarly many A4 -> A5 etc etc. So the target may be generated by a series of successive expansions following each rule with inserted terminal. But since we cannot create new rules but we can insert terminals anywhere, we could insert a terminal in each of those rules to match the target.

But the target length is 42; we already have A78 covering the first 2 terminals. There are many other expansions that could be used to produce each subsequent terminal individually, like each A4 -> A5 may produce "1". But we need different numbers.

However, many productions have other terminals built-in. Possibly we can use some expansions where terminal numbers appear directly in the RHS. For example, A4 -> A5 (no terminal). A4 -> A18 (no terminal). A4 -> A19 (no terminal). A4 -> A20 (no terminal). So by default those expansions produce no terminals unless its sub-nonterminals produce them. A5->1 gives 1; A18->? it has production A18->1, and also A18->A15 A20 etc. So the only explicit terminals are terminal 1. So using A4 expansions without extra inserted terminals yields only 1's maybe.

Thus to generate numbers other than 1, we must insert them somewhere in those productions. For example, we could insert 34 into a rule: maybe in B: A4 -> A5, we could modify to A4 -> 34 A5 or A4 -> A5 34 etc. Similarly in A5 productions we could insert terminals.

But to minimize insertions, likely we should insert the needed non-1 terminals directly into those many identical rules for A2 -> A78 A3, but that seems not helpful as that rule only used once per A2 expansion (which we use only once). Similarly we could use many expansions of A4 -> A5 as many times as we need to produce a chain of numbers, each time inserting an appropriate integer before or after A5. However each insertion counts toward total inserted terminals.

Thus we need to compute minimal number: maybe we could use a single rule that already contains a long sequence matching most of S. For example, A9 -> 34 31 A25 13 32 has 5 specific numbers matching prefix and suffix.

Let's examine S in segments:

S: [34,31,14,33,29,28,27,26,1,13,32,30,31,25,...]

In the prefix, we have 34,31. Then later we have 14,33, then after a series of 29,28,27,26,1,13,32,30,31 => maybe many can be derived by other expansions.

Observe the numbers 34 and 31 appear consecutively at start. They also appear later in S after some numbers: 30,31 is at index 14? Actually index 14 is 30, index 15 is 31. So we have another 31 after 30.

Also later we have 25,21,16,24,... and among these 25 appears at index 16? Actually target: after ...30,31, then 25 at index 16, then 21,16,... and later we have 12,8,9,10,3,5,4,2,... and final 1,7,11,6,8,1.

Thus maybe there are nonterminals that generate those subsegments.

Let's list known productions that contain these subsegments:

- A9 -> 34 31 A25 13 32 (covers 34,31, then some A25 yields maybe a sequence, then 13,32). Our S after 34,31 is 14 (not part of A9). But perhaps we can insert 14 after 31 before A25, thus modify rule_351 to include 14.

But we also have separate rule for 33? Possibly we could insert 33 after 14.

Alternatively, there is a production A49 -> 34 14 A25 32 30, which contains 34,14, (some A25), 32,30. In S after 34 we have 31 then 14 then 33 then 29 etc. So A49 doesn't match 31.

But there is a production A49 -> 34 14 33 A25 13 31 (includes 34,14,33,13,31). That's close but missing 31 after 34.

Actually A49 -> 34 14 33 A25 13 31 yields: 34, 14, 33, <A25>, 13,31. Our target prefix: 34,31,14,33,29... So we need 31 immediate after 34, then 14,33. In A49 rule we have 14,33 then later 31. So we can insert 31 after 34 and before 14, maybe.

Goal: Use a rule that already contains many numbers and insert the missing ones.

Another candidate: A66 -> 34 14 33 A25 13 32 contains 34,14,33,13,32. The target after 34,31,14,33,... then later 13 appears after 1, then after 32 preceded by 13? Actually we have 34,31,14,33,... later we have 13 at position 11 after the series of numbers. So not immediate.

Thus there may be multiple insertion steps.

Now we need to produce the entire S using minimal terminal insertions. A rough heuristic: use a production that contains many numbers already matching the target, to minimize insertions.

Let's examine A9 productions:

- rule_351: "34 31 A25 13 32". Numbers present: 34,31,13,32. Missing numbers from S: after 34,31 we need 14,33,... etc. So this covers 34,31 at start, and later 13,32 close to later in sequence (index 10 and 13?). Indeed in S, after many numbers (including 1 at index 9) we have 13 at position 10, then 32 at 13. The relative order is 13 appears before 32, but there are other numbers between them in S: 34 31 14 33 29 28 27 26 1 13 32 ... Actually 13 appears just after 1, and 32 appears after 13 (immediately at position 13). So S's segment "13 32" appears consecutively at positions 10 and 13? Wait indices: Let's recount S with positions:

1: 34
2: 31
3: 14
4: 33
5: 29
6: 28
7: 27
8: 26
9: 1
10: 13
11: 32
12: 30
13: 31
14: 25
15: 21
16: 16
... Wait I mis-index earlier. Let's produce complete S with correct positions:

S positions (starting from 1 for the suffix after initial two 1's):
1: 34
2: 31
3: 14
4: 33
5: 29
6: 28
7: 27
8: 26
9: 1
10: 13
11: 32
12: 30
13: 31
14: 25
15: 21
16: 16
17: 24
18: 19
19: 17
20: 18
21: 20
22: 1
23: 15
24: 23
25: 22
26: 21
27: 12
28: 8
29: 9
30: 10
31: 3
32: 5
33: 4
34: 2
35: 1
36: 7
37: 11
38: 6
39: 8
40: 1

Now we see segment: positions 9-11: 1 13 32. Indeed after 1,13 then 32. That's exactly "1 13 32". The rule A9 -> 34 31 A25 13 32 yields 34,31, then something from A25, then 13,32. So after A25 we have 13,32. That matches positions after some segment up to the 1 maybe. So the A25 may generate the segment: 14 33 29 28 27 26 1? Wait we need to align.

Goal: produce S = 34 31 [X] 13 32... So maybe we can use A9 as root for A3 (or nested). Let's hypothesize: A3 -> A27? Actually A27 may produce an A28 etc that eventually lead to A9? Not directly. But we could decide to produce S using A3 -> A44 which may produce A45 expansions. Or use A3 -> A61 (like A62-> A66 -> A9...). Let's see: A66 is reachable from A62 which is reachable from A61 which is reachable from A3. So A3 -> A61 -> A62 -> A66. Then A66 could use rule 1387 to produce "34 14 33 A25 13 32". However we need "34 31 ..." not "34 14". So maybe we can insert terminal 31 after 34 in rule_1387. So modifications: rule_1387 currently "A66 -> 34 14 33 A25 13 32". Insert 31 between 34 and 14: new RHS "34 31 14 33 A25 13 32". That would produce 34,31,14,33,... 13,32 exactly matching target up to 32 after A25.

Now we need to generate the segment after 34,31,14,33 which is "29 28 27 26 1" before we get to 13. Wait in target we have after 33: 29 28 27 26 1 then 13 32 ... In our new A66 RHS, after "33" we have A25 which could produce "29 28 27 26 1"? Let's check what A25 yields. A25 has productions:

- rule_422: A25 -> 1
- rule_423: A25 -> A26 A25

Thus A25 can produce sequences of numbers from repeated A26 expansions followed by 1? Actually A26 produces terminals 26,27,28,29 via rules 424-427: A26 -> 26,27,28,29. So A25 can recursively expand: each recursion adds an A26 (one number) then recursively call A25 again, then finally a 1. So A25 can generate any sequence of numbers from {26,27,28,29} in any order? Since the recursion always adds numbers from A26 at the front (A25 -> A26 A25). So the sequence will be a series of numbers from A26 (each chosen from the 4 terminals) followed by a final 1. It cannot produce 29 then 28 then 27 then 26 then 1? Actually each step chooses one of A26's productions. So we can get any order by picking appropriate terminal each step: For example for sequence: 29 28 27 26 1 we can produce:

A25 -> A26 A25 (choose 29)
then inner A25 -> A26 A25 (choose 28)
inner -> A26 A25 (choose 27)
inner -> A26 A25 (choose 26)
inner -> 1.

Thus we can generate exactly 29 28 27 26 1. So this matches our needed segment after 33.

Thus by using modified rule_1387, we can generate prefix 34,31,14,33, then A25 generates "29 28 27 26 1", then we have terminal 13,32 as per rule.

Thus the start of S up to position 13 (34,31,14,33,29,28,27,26,1,13,32) are covered. After 32, the target includes 30,31,...

Now after 32, we have 30,31. In our rule after A25 we have terminals "13 32". So after that, we need to produce 30,31. The continuation after A66's RHS (after "13 32") will derive from the rest of A66's derivations? That is, after the RHS ends, we might have further expansions via recursion of A66 (since there is rule_1386: A66 -> A63 A66, and rule_1390: A66 -> A66 A65). However, those recursions will place other nonterminals after the RHS. So we could use recursion to generate 30,31 later.

Specifically, we could use the rule A66 -> A66 A65 for recursion at the right side, i.e., after the main sequence "34 31 14 33 A25 13 32", we could have A66 produce further symbols via A66 A65. But that seems to create duplicate preceding nonterminal (it recurses on left side). Actually A66 -> A66 A65 means it expands to A66 (again) followed by A65. So that results in left recursion: leads to infinite loops but we can stop after some expansions, using A66 base rule. To generate suffix after the base, we need a rule that appends something after the base. That can be done by A66 -> A66 A65 (the A65 will derive something next). So after base rule (the one we modified), we can have a chain of A66 -> A66 A65 -> A66 A65 A65 ... etc. But the base case that does not contain recursion is maybe the rule we inserted (modified rule_1387). Adding later expansions may add extra numbers after 32 from the sequence generated by A65 (or A63 etc). Let's explore A65.

A65 has productions:

- rule_1382: A65 -> A64 A66
- rule_1385: A65 -> A65 A65
- rule_1384: A65 -> A71 A67
- rule_1385 actually duplicates? Let's list:

From earlier:

A64 -> 1 (rule_1376)
A64 -> A64 A63 (rule_1377)
A64 -> A65 A64 (rule_1378)
A64 -> A71 A68 (rule_1379)
A64 -> A76 A73 (rule_1380)

But we need A65:

rule_1382: A65 -> A64 A66
rule_1383: A65 -> A65 A65
rule_1384: A65 -> A71 A67
rule_1385: A65 -> A76 A74

Wait the rule numbers I need to verify: The listing shows:

- rule_1382: A65 -> A64 A66
- rule_1383: A65 -> A65 A65
- rule_1384: A65 -> A71 A67
- rule_1385: A65 -> A76 A74

Thus A65 can produce terminal sequences via many expansions.

But we need to generate after the prefix: 30,31,25,21,16...

How might these be generated? Perhaps using A63 (or A63) which may produce numbers like 30,31 etc.

Check A63 productions: rule_1371: A63 -> 1; rule_1372: A63 -> A62 A73; rule_1373: A63 -> A63 A63; rule_1374: A63 -> A66 A64; rule_1375: A63 -> A70 A68.

But A63 does not directly have numeric terminals besides 1. So not helpful.

Let's look at A71: rule_1412: A71 -> A64 A70; rule_1413: A71 -> 25 A23 22 (produces 25, then maybe later). rule_1414: A71 -> 25 A23 23.

Thus A71 can generate "25 22" or "25 23". In target, after 30,31 we have 25,21,16,... So not "25 22". Could be other.

But note that rule_1415: A71 -> A64 A70; rule_1416: A71 -> A65 A71; rule_1417: A71 -> A71 A69; rule_1418: A71 -> A76 A72.

Thus A71 can recursively generate sequences.

Now investigate A70 and A69 etc.

A70: can produce A62 A72, A63 A70, A66 A71, A70 A69.

So chain continues.

Seems extremely complicated.

However, maybe there is an easier path: Use A4 expansions (since A3 can go => A4) and then A4 -> A20 which includes sub-sequences: A20 is defined with many productions that include terminal numbers like "12 9 10 A21 8" etc. Also includes A20 -> A10 A19 (but A10 and A19 may produce various numbers). But A20 also has a production "A20 -> 12 9 10 A21 8". This includes terminal 12,9,10, then A21 (which expands to A22 (2-5) recursively), then 8. This sequence contains 12,9,10, then maybe some numbers from A22 (some of 2-5), then 8. In our target, after the earlier big prefix we have 12,8,9,10? Actually target includes after 21 (position 27?), we have 12,8,9,10 (positions 27-30). Indeed after position 26 we have 21 (index 26), then 12 (27),8 (28),9 (29),10 (30). So we need that after 21.

Thus perhaps after generating prefix up to 21 we can use A20 production "12 9 10 A21 8". But order is 12 9 10 *8* but target order is 12 8 9 10. Slightly different: target: 12,8,9,10. The rule has 12,9,10,...8. So we might need to insert terminals to reorder? Wait we cannot reorder; we can only insert. So we cannot change 9,10 order relative to 8. But we can insert extra 8 after 10, but target has 8 before 9,10; we can also have extra terminals before that produce the needed 8 earlier then maybe A21 producing something else that includes 8? Actually A21 expansion yields A22 something etc; but not 8.

But there is also rule_1445: A43 -> 12 8 9 10 A21 (notice order 12 8 9 10 ...). This more matches target: 12 8 9 10 then A21. Good! A43 yields after those numbers A21. Then A21 can produce sequence of numbers 2-5 maybe, but we need 3 5 4 2 after that. Indeed target after 10 is 3,5,4,2, then 1 (position 35). So if we use A43 -> 12 8 9 10 A21, then A21 must generate 3 5 4 2 1? Let's examine A21 productions:

- rule_410: A21 -> 1
- rule_411: A21 -> A22 A21

Thus A21 can generate a sequence of terminals: any number of A22 (which yields a single integer 2-5) followed by recursively more A21, finally terminating with 1. So A21 can generate any sequence consisting of any number of 2-5 terminals followed by 1. The sequence we need after 10 is 3,5,4,2,1 (positions 31-35). That matches: 3,5,4,2,1 are all in {2-5} with final 1. Thus we can generate exactly 3 5 4 2 1 using A21: Choose A22->3, then A21 -> A22 A21, choose A22->5, then A21 -> A22 A21, choose A22->4, then... etc. That's possible. So A43 -> 12 8 9 10 A21 yields the correct order: after 10, we derive A21 that yields 3,5,4,2,1. That matches target exactly.

Thus A43 is a great candidate for the suffix. Where is A43 reachable from? A3 -> A44 (rule_68) is one path. A44 can produce many A45 expansions (rules_768-... etc). Eventually A45 expansions may lead to A43 perhaps via other nonterminals (A45->?), but perhaps A44 includes rule 832 to rule 895 that indicates A44 -> A58. Later A58 expansions may lead to A43 via A44 -> A58? Actually rule_832: A44 -> A58 and many repeats. So A44 can produce A58.

Now A58 production (rule_1093) expands to 1, A55 A60, A56 A45, A57 A59, A58 A58. So A58 can produce A55 A60, and A55 can expand to ... and A60 expansions include many numbers perhaps. But A58 does not directly produce A43.

Alternatively, A44 can directly produce A58, but not A43. There's also possibility that rule_768-... for A44 only produce A45, and A45 in turn can produce ... eventually leads to A43? Let's examine A45 productions (rules_1024-1028). A45 -> 1; A45 -> A45 A58; A45 -> A46 A45; A45 -> A49 A59; A45 -> A53 A60. So A45 can produce A53 A60 eventually; A53 expansions include some A45 A55 etc. Not directly to A43.

But A44 also has rule_832 to A58, maybe A58 can produce A60 then maybe A60 leads to A43? Actually A60 productions: rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 (lots). Also rule_1104 includes A21 and also numbers 7,11,6,8 after that. Not directly A43. There's also rule_1105: A60 -> 12 9 A21 6; rule_1101: A60 -> A59 A58; rule_1102: A60 -> A60 A58 (recursive). So A60 may generate sequences of numbers, but not A43.

We might need to use A43 directly reachable from A3: Indeed A3 -> A44 (rule_68). Then we could have A44 -> A45? Wait A44->A45 (lots). Then A45 may produce A45 A58 etc. Not straightforward.

Alternatively, we could choose A3 -> A27 or A61 etc that can produce A43? Let's examine A27 expansions: rule_428-... A27->A28 is repeated many times; also some expansions go to A41 (rule_492 onward). But no direct A43.

The path to A43 may be via A61 -> A75? Actually A61 goes to A75, A76, etc. These eventually produce A... but not A43.

But maybe we can produce A43 using A4 expansions: For A4 we have rule_134: A4 -> A18; though A18 doesn't produce A43.

But note that A4 is from A3 -> A4 (rule_66). So we could go A3->A4 then A4->A18 or A20 etc. Since A20 can produce A43 via some path? Actually A20 does not produce A43.

But A4 -> A20 (rule_262 onward). A20 expansions may lead to A21 or A59 etc. But A20 does not produce A43 directly.

But we might also keep A3->A44 eventually leading to A43 somewhere else.

Let's systematically explore reachable potentials:

From A3 we have:

- A4
- A27
- A44
- A61

Goal S's first part (prefix) is 34 31 14 33 etc. Already we found a candidate: use A66 reachable via path A3->A61->A62->A66. That covers prefix up to 32. The remaining suffix (starting at 30 31 ...) may then be derived by recursive expansions from A66 or deeper.

However we also need later segment ... 12 8 9 10 3 5 4 2 ... The A66 recursion may not lead directly to A43. But perhaps we could finish using A66 recursion to produce the suffix via other rules (like using A65 -> A71 A67 etc). Yet this may be complex.

Another approach: perhaps after A66 produced prefix up to 32, we can have A65 generate 30,31,... and later suffix with A43 by using other expansions.

Let's explore A65 chain for generating 30,31.

A65 -> A64 A66 (rule 1382). That adds A64 before A66 (not after). But we might use recursion in A66 -> A66 A65 to append after. So after base A66, we get A66 A65. The new A65 could produce some numbers and then maybe more A66 recursion etc.

Thus we could generate suffix after base by using A66 recursion: A66 (base) -> [prefixSequence] then follow with A65 (via A66 -> A66 A65). The new A65 can use rule A65 -> A71 A67, or maybe other to generate numbers.

A71 produces 25,22 or 25,23; plus other recursion might yield numbers after 25. The target after 32 is 30 31 25 21 16 ... So we need to generate 30 31 after 32, then 25 etc.

Now where can we get 30? Possibly from A70->? but not immediate. Let's examine productions that produce 30 directly as terminal: rule_327: A5 -> A5 A18 includes no 30. Rule_352: A10 -> 1 (no). But actual numbers appear in some nonterminals: A21 has only 1 and numbers 2-5. A22 has terminals 2,3,4,5. A23 has 1. A24 has 17,18,19,20. A25 -> A26 A25: A26 outputs 26,27,28,29. A26 -> 30? Actually not: A26 possible terminals are 26,27,28,29. No 30. A27? not. A28? its productions include A28 -> A28 A41 etc., but no terminals. However there is rule_698: A30 -> A42 A39 etc. Not direct. But there are productions with terminal 30, like rule_322: A4 -> A20 (makes A20). But A20 has production containing "12 9 10 A21 8". Not 30.

But there is rule_326: A5 -> 1, not 30. rule_334: A6 -> A9 A7; A9 includes production maybe producing 34 31 A25 13 32, not containing 30. But there is rule_351: A9 -> 34 31 A25 13 32 includes A25 which can produce 26-29 but not 30. However A25 can only yield numbers 26-29 and final 1. It cannot produce 30 directly. So maybe 30 is generated through other productions like A20 -> 12 9 10 A21 8 (includes 12,9,10,8). Not 30. However rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 includes 12,8,9,10,...,7,11,6,8. Not 30.

But there is rule_698: A30 -> A42 A39 (but A30 may also produce some numeric terminals? No direct numbers.

A31 production etc. Some productions contain 30 as terminal, e.g., rule_706: A32 -> A29 A32 (no). However rule_152 etc? Actually rule_701: A31 -> A31 A31 etc. Not.

But maybe 30 appears in rule_326? That's 1. No.

Maybe 30 appears embedded as part of rules generating 30 via other sequences like A31->... but perhaps not.

Search for "30" as terminal in the production rules: I saw some occurrences:

- rule_298: A4 -> A20 (no)
- rule_322? Not given.
- rule_311: maybe A33->A34 etc.

Let's search manually in the provided text:

We have a line: rule_322: "A4 -> A20" doesn't have 30. At rule_324: A4 -> A5 (only.

But scanning: There's rule_322: Actually it's near the middle: rule_321? Let's find "30" occurrences:

- rule_326: A5 -> 1
- rule_331: A6 -> 1
- rule_332: A6 -> A5 A16
- etc.

Scrolling down: rule_351: A9 -> 34 31 A25 13 32 has 34,31,13,32.

Later: rule_408: A20 -> 12 A21 6 (includes 12 and 6).

rule_406: A20 -> 12 9 10 A21 8 (includes 12,9,10,8).

rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 (includes 12,8,9,10,7,11,6,8).

rule_1445: A43 -> 12 8 9 10 A21 (includes 12,8,9,10).

rule_1446: A43 -> 12 A21 (includes 12, A21).

rule_1440? Not.

Also rule_220? Not.

But terminal 30 appears where else? Wait I recall grammar may have a rule: A4 -> A20 (and A20 may produce 30 through something else?). Let's check A20 productions: list:

- rule_403: A20 -> 1
- rule_404: A20 -> A10 A19
- rule_405: A20 -> A11 A5
- rule_406: A20 -> 12 9 10 A21 8
- rule_407: A20 -> A12 A20
- rule_408: A20 -> 12 A21 6
- rule_409: A20 -> A20 A18

Thus none include explicit 30. However note that A10 gives 1 or other expansions; A19 maybe yields numbers etc. Not directly 30.

But there is rule_382: A15 -> A18 A15 (recursive) and also A15 -> A15 A12 etc.

Thus to get terminal 30, we need a rule that directly includes 30, maybe through A8 or elsewhere. Let's search more thoroughly: Look at rule numbers beyond 1000: Not many.

Search for "30" likely appears in many A4 -> A20? Actually A20 is numeric expansions maybe produce 30 via recursion with other numbers.

But I recall rule_698: A30 appears as nonterminal not terminal.

Thus perhaps terminal 30 never appears as a literal in the original grammar. However target includes 30. That suggests we must insert terminal 30 somewhere into a production rule. Indeed, we need to add at least one insertion for 30 because no rule currently contains it.

Similarly, terminal 33 appears in some rule: A66 includes 33; also maybe A9? It does not include 33; A66 includes 33. So 33 is present. Terminal 29 appears only as A26 -> 29. So we have that. Terminal 28 -> A26 -> 28. Terminal 27 -> A26 -> 27. Terminal 26 -> A26 -> 26. Terminal 31 appears as a terminal in A9 (34 31 ...). Also appears maybe in some other rules.

Terminal 25 appears as A71 (provides 25), also as A71-> 25 A23 22 etc. And also as part of rule_698? Not needed.

Terminal 21 appears as A22->? Actually A22 includes 2,3,4,5 only. 21 appears in some productions: possibly as terminal 21 in rule_1544: A85 -> 21. Also there are A71 -> 25 A23 22 (no 21). But we have terminal 21 as part of some productions? There is rule_1545: A85 -> 22. So 21 appears from A85 only.

Terminal 16 appears? I haven't seen a direct 16 terminal. Look at A22 productions: no 16. A23 is 1. A24 includes 17-20. A31? no. A32? maybe includes "34 33 A25 13 32 30 31" includes 30,31, but not 16. Possibly 16 appears as terminal from A22? Not. But perhaps rule_752: A41 -> A38 A43 includes A38 maybe leads to 16? A38 expansions: rule_738: A38 -> A39 A36; not include 16. Actually there is rule_438 maybe, but not.

Thus many target terminals may be missing from the grammar and need to be inserted.

We need to determine minimal set. Since we can insert any number of terminals directly into production rules, we could insert all missing terminals into a single rule used in derivation.

Potential strategy: Use simple derivation where we start with A1 -> A2 -> A78 A3. A78 we use to produce the first two '1's via A80 A79. A3 we then could produce a single production rule that yields the whole suffix using inserted terminals.

We could modify rule for A3 (say rule_66: A3 -> A4) to insert the entire remaining sequence after A4, as "A4 <terminals>"? Wait we cannot insert after nonterminal? Actually we can insert terminal anywhere in RHS: e.g., RHS currently "A4", we could change to "A4 34 31 14 ..." but then seems we need A4 to produce nothing or produce other terminals. But A4 expansions produce many sub-terminals; we can use A4 -> A5 (which yields 1). But we don't want extra 1's in prefix of suffix; we want next token after "1 1" to be 34. So we could avoid using A4 or make A4 produce epsilon? Not possible. So maybe we should use a different rule for A3: Choose rule_68: A3 -> A44. Replace A44's productions to produce the suffix.

Alternatively, we could directly modify rule_67: A3 -> A27. A27 can derive maybe A28 etc and eventually we could insert terminal sequence.

But maybe we can create a rule for A3 that directly yields the suffix via insertion: e.g., rule_66 is A3 -> A4. Instead, we could keep A4 as a nonterminal that we can add inserted terminals after it (or before). But A4 again must produce something maybe only 1's... But if we also modify A4's productions to directly include the suffix numbers (by insertion), we can absorb the entire sequence.

But the rule restrictions: we cannot delete or reorder symbols. So we can only add terminals. Changing a rule's RHS from "A4" to "K A4 L" where we insert terminals before/after A4 (i.e., "34 31 14 ... A4" or "A4 34 31 ..."). However, the order is crucial: original order is just A4. So we can insert terminals before or after A4, but both sides are allowed? The rule is "A3 -> A4". Inserting terminals anywhere in existing RHS means we can hang before A4, after A4, or in between non-terminals (but there are no between). So we could modify it to "A3 -> 34 31 14 ... A4" which would produce the suffix plus whatever A4 generates. Since A4 can generate maybe empty? No epsilon. So that would add extra. Perhaps we could choose to modify A4's productions to become epsilon? Not allowed to delete symbols; but we could insert terminals, but not remove A5 etc. For A4 we have many productions like "A4 -> A5". If we insert after the A5 a huge terminal sequence that yields the rest after A5? That makes extra A5's contributions; but we could set A5 -> 1 and then after A5 add the rest terminals. That would produce "1 <suffix>" which adds an unwanted 1 before the rest. But if we insert additional terminals before A5, we could maybe produce suffix starting with those inserted terminals before the 1. Considering we need the suffix to start with 34 (not 1), we could modify A4 -> (suffix) A5 or A4 -> A5 (suffix after). But whichever we choose, we will have an extra 1 at either beginning or end. If we put suffix before A5, we get "34 ... A5", and A5 yields "1". So the trailing 1 would be after the suffix. Actually target ends with 1, that might be perfect: suffix ends with ... 8 1, and we could have last A5 produce that final 1. Let's check: The target ends with "8 1". So if we make A4 -> (suffix without its final 1) A5 with A5->1, we could produce suffix ending in 1 correctly. But the suffix after the 34 prefix goes all the way to the final 1. So we could structure: A3 -> A4; modify A4's rule to have huge sequence of terminals (the entire suffix) before A5 (or after A5) with appropriate placement of final 1.

Idea: Use a single A4 production, e.g., rule_70: A4 -> A5 (original). We'll modify it to "A4 -> <allRemainingTerminalsExceptLast1> A5". Since A5 produces "1", we get full suffix ending in "1". Then A5's 1 corresponds to final 1 in target. However we need to ensure no other productions for A4 conflict: there are many other A4 -> A5 rules (duplicate). But we can modify only one of them (or some). However the derivation may choose any of the numerous A4 -> A5 rules; they are identical. So if we modify one to include the big suffix, the others remain unchanged and still produce just A5. Since the parser (or derivation) may use the modified rule, we can select it. Thus minimal insertions: we need to insert all 40 terminal symbols of the suffix into that rule (maybe with a few extra inserted terminals for alignment). That's 40 inserts, which may be minimal if there are fewer missing terminals across the whole grammar? Maybe we could use existing terminal sequences already present to reduce insert count: e.g., we can use rule_1387 inserted earlier containing many terminal numbers (so many of the suffix already covered). This could reduce demand for multiple insertions across many productions.

Thus the problem reduces to: we need to produce S using the grammar, but we can modify many rules. To minimize insertions, we should use existing productions that already contain many of the needed terminals. For example, rule_1387 includes 34,14,33,13,32, but missing 31 and 29,28,27,26,1. However we can insert 31 after 34 and 29,28,27,26,1 before 13, maybe using A25's recursion to generate those numbers (so we don't need to insert those numbers). So we only need to insert 31 after 34 and maybe 30 and 31 and 25 etc later. That seems fewer than inserting all numbers.

Thus we see the benefit of using existing productions like the A66 rule.

Now we need to generate after 32 the rest: 30 31 25 21 16 24 19 17 18 20 1 15 23 22 21 12 8 9 10 3 5 4 2 1 7 11 6 8 1.

We could perhaps use A20 -> 12 8 9 10 A21 (rule_1445) to generate the 12 8 9 10  etc segment, and also generate 3 5 4 2 1 afterwards via A21. Good.

Now what about the segment before: after 32 we need "30 31 25 21 16 24 19 17 18 20 1 15 23 22 21". Perhaps we can use other expansions.

I suspect there is a production for "30 31" maybe in rule_708: A32 -> 34 33 A25 13 32 30 31 includes "30 31" after the other part. This rule is A32 -> 34 33 A25 13 32 30 31. Actually rule_708: A32 -> 34 33 A25 13 32 30 31. Include numbers 34,33, A25,13,32,30,31.

If we could use A32 somewhere after A66's prefix that already gave us 34,31,14,33,... but maybe the second occurrence of 34,33 is not needed.

Alternatively, maybe there is a production for 30 31 without others.

But exploring deeper: A30 -> A30 A29 etc; not directly.

But rule_152? No.

However, rule_152: from A3? No.

Alternatively, rule_361: A11 -> A20 A16 includes A20, which can produce 12 8 9 10 A21 etc but not 30.

But A16->A15 A11 includes recursion.

It's quite complex. But maybe we can piece the suffix using two major blocks: A66 block for prefix up to 32; A43 block for the latter part from 12 onward; and we need to generate intermediate part (30 31 25 21 16 24 19 17 18 20 1 15 23 22 21) some of which may be produced by other rules.

We could treat that block using A18 or A19 expansions or A20 expansions? Let's check A18: productions include 1, and also A15 A20 (which may involve A20). So maybe using A18 -> A15 A20 can generate something?

But A15 includes productions that have numbers: A15 -> 1; A15 -> A15 A12; A15 -> A16 A13; A15 -> A17 A14; A15 -> A18 A15. Not many terminals.

Thus, maybe using A70 etc.

But given the heavy freedom, we may consider a simpler greedy solution: Insert the entire missing suffix into a rule where it's easy to place and generate no extra terminals. For example, we could modify rule_66 (A3 -> A4) to "A3 -> A4 34 31 14 33 ... 1", and then use A4 -> 1 (via A5 -> 1 etc) for the trailing 1? Not needed maybe.

Alternatively, we could modify rule_1391: which is "A66 -> A70 A67"? No.

But maybe easiest: Use A2 -> A78 A3 and modify it to include the entire suffix after A3: "A2 -> A78 A3 <suffix>" where suffix includes all remaining tokens. But again after A2 we still have A78 produce "1 1". So we could place suffix after A3. Then A3 derived from A4->... maybe just produce epsilon? No. But A3 must be derived eventually to something that yields at least something; we could keep A3 -> epsilon if there were an epsilon rule, but none exist. So we cannot just ignore A3.

Thus we need to use A3 and produce minimal terminals through it; maybe just produce the final "1", and all other tokens we can insert in A2 rule after A3: "A2 -> A78 A3 <rest>" where "rest" is our suffix, but again we need A3 to generate some tokens (maybe just empty). However A3 must derive something, but it cannot be empty; but we could set A3 -> A5 where A5 -> 1, and then after A3 we insert the rest of the suffix (including maybe the final 1?). This would produce "1 1 1 <rest>", i.e., extra 1 at some position. We could potentially insert a terminal before A5 to offset. But seems messy.

Better approach: Use A4's many productions to generate extra tokens via recursion. Since many repeated rules for A4 -> A5, A4 -> A18, etc, we could possibly chain them to get sequences of numbers with recursion using other nonterminals. But we can also just insert needed tokens in the appropriate rules.

Given we need a feasible solution but also minimize insertions. However there may be many missing terminals, but we can use a small number of new productions by leveraging existing ones.

Given the difficulty to compute the absolute minimum, but we need to provide some updated production rules (the inserted terminals) as answer. The problem may be expecting a simple solution: insert the missing terminals into a specific rule (e.g., rule_1387) and maybe into rule_708 and rule_1445 etc.

The typical "minimum insertion" might be to insert 31 after 34 in rule_1387 and to insert "30 31" after 32 using rule_708? But it's tricky.

Maybe we can find a more systematic approach: Use this derivation:

Start A1 -> A2 -> A78 A3

A78 -> A80 A79 => 1 1

Now for A3, we choose A3 -> A44 (rule_68). Then we need to produce suffix S from A44.

A44 has many productions to A45 (and to A58 later). Perhaps A45 is a good candidate to generate numbers via A49->... etc.

Specifically, A45 can go to A49 A59 (rule_1028). A49 has "34 14 A25 32 30" etc. Might help produce prefix.

Let's examine A49 alternatives:

- rule_1045: A49 -> 34 14 A25 32 30 (numbers 34,14, then A25 yields 26-29, then 32,30). This yields prefix 34 14 {seq of 26-29} 32 30. But we need 34 31 14 33 etc. Not perfect.

- rule_1046: A49 -> 34 14 33 A25 13 31. (34,14,33, then A25,13,31). This covers 34,14,33,... and 31 at end after A25. But we need 34 31 14 33 after A25? Not match.

- rule_1044: A49 -> 1 (just 1).

- rule_1047: A49 -> 34 A25 30 (contains 34 then A25 then 30). That could produce prefix 34,A25,30. But need 31 after 34.

Nonetheless we can insert missing numbers into a rule. For prefix: we need 34 31 14 33 after A80 A79 (so after first two 1's). Suppose we use A45 -> A49 A59 (via rule_1028), then modify A49's RHS to produce "34 31 14 33 A25 13 32". Wait we could use rule_1046 and add "31" after 34.

But we also need A59 after that to produce the rest? Actually A59 expands to various numbers including 1 etc; but maybe not needed much.

Alternatively, we could use A49 -> 34 31 14 33 A25 13 32 directly (by inserting 31 after 34). Then we need to generate "30 31 ..." after that, but also the rest.

But rule_1046 already includes 13 31 at end; we might need to push things.

Let's see possible path: A45 -> A49 A59. Let A49 produce the prefix up to 32 inclusive. Then A59 can produce the rest of suffix.

Now A59 expansions: rule_1098: A59 -> 1; rule_1099: A59 -> A47 A45; rule_1100: A59 -> A48 A59; rule_1101: A59 -> A54 A60; rule_1102: A59 -> A59 A58. So A59 can produce a lot via recursion and A60 which includes numbers 12 8 9 10 etc. So A59 may be used to generate the later segment (from 30 onward). For example, we could use A59 -> A54 A60; A54 -> 1 ... but eventually maybe produce numbers like 30 etc? Not clearly.

But perhaps A59 -> A54 A60, where A54 production includes terminal 25 21 ... via rule_1074? Actually A54 can produce A47 A53 etc, but also rule_1073: A54 -> 25 21 24 A23 15 23 22 (contains numbers 25,21,...). This matches part of our suffix after 30 31. Indeed the segment after 30 31 is: 25 21 16 24 19 17 18 20 1 15 23 22 21 ... Actually we have 25,21 as first two numbers there. A54 can produce "25 21 24 A23 15 23 22". Slightly off: we need "25 21 16 24 ..." but we get 24 then A23 then 15 then 23 then 22. Not quite.

Let's examine A54 more precisely:

- rule_1073: A54 -> 25 21 24 A23 15 23 22

Thus yields sequence: 25,21,24, then derivation of A23 (which yields 1), then 15,23,22.

Our target after "30 31" begins with "25 21 16 24 ...". So we have extra 16 after 21, before 24.

Thus we could modify that rule to insert "16" after 21. That would give 25 21 16 24 ... which matches.

Thus A54 can generate part of suffix! Good.

Now after A54's sequence, we need "19 17 18 20 1 15 23 22 21 ...". A54's rule includes "A23" (which yields 1) after 24. Then "15 23 22", giving 15,23,22. That yields 1 15 23 22; we have extra 19 17 18 20 and then 1 15 23 22. So maybe A60 or other expansions can supply the missing numbers 19 17 18 20. Also later after 22 we need "21 12 8 9 10 3 5 4 2 1 7 11 6 8 1". This later part can be generated from A60 and A21 etc.

Thus overall architecture could be:

- A49 with inserted 31 after 34 yields part: "34 31 14 33 A25 13 32". After A25 produces "29 28 27 26 1", we have "13 32". End of A49. Actually we may want after that immediate "30 31" before moving to A54. So maybe we can make A25 produce end "29 28 27 26 1 30 31"? But A25 only produces 26-29 and final 1. Cannot produce 30 31. So we need to insert "30 31" after A25 or after 32.

Thus after A49 produces "...13 32", we then need to have "30 31". We can achieve that by inserting "30 31" after A49's RHS (i.e., before linking to A59). So modify rule_1028 (A45 -> A49 A59) by inserting terminals "30 31" between A49 and A59: A45 -> A49 30 31 A59. This adds two terminals.

Now after that, A59 will generate remaining suffix: starting with 25 21 16 24 19 17 18 20 1 15 23 22 21 12 8 9 10 3 5 4 2 1 7 11 6 8 1 (still need to get 19 17 18 20 and later pieces). So we need A59 to generate that.

We can use A59 -> A54 A60 (rule_1101) to generate 25...22 plus later part (via A60). That gives A54's sequence: 25 21 24 1 15 23 22, but missing 16 and the segment 19 17 18 20 before that maybe. Actually we earlier inserted 16 after 21 via rule_1073, giving: 25 21 16 24 1 15 23 22 (still missing 19 17 18 20). But we need 19 17 18 20 after 16? Wait target: after 16, we have 24,19,17,18,20 (then 1 via A23). Actually target segment after "25 21 16 24" is "19 17 18 20". So we need to generate 19 17 18 20 between 24 and the 1. In A54's rule we have after 24 -> A23 (1). There's no place for 19 17 18 20. We could modify the rule by inserting "19 17 18 20" after 24 and before A23: modify rule_1073 to "A54 -> 25 21 16 24 19 17 18 20 A23 15 23 22". That would embed those numbers. So we need to insert 16 (if not already inserted), 19,17,18,20 (which are four numbers). However we might also need to adjust to have correct ordering.

Original rule_1073: 25 21 24 A23 15 23 22. We'll now change to insert after 21 a 16, after 24 insert 19 17 18 20. So total inserted terminals: 16, 19, 17, 18, 20 => 5 inserted terminals.

Now after the 1 from A23, the rest "15 23 22" are present and match target after the following 1. Good.

Thus we have covered "25 21 16 24 19 17 18 20 1 15 23 22". Good.

After this, we need "21 12 8 9 10 3 5 4 2 1 7 11 6 8 1". That can be done via A60 (includes 12 8 9 10 ...) and further A21 for the rest.

Specifically, A60 -> 12 8 9 10 A21 (rule_1445) yields 12,8,9,10 then A21 yields sequence of numbers 2-5 ending with 1. However the target after "22" is "21 12 8 9 10 3 5 4 2 1 7 11 6 8 1". So we need a leading 21 before A60's block. Then after A60 we need "7 11 6 8 1" after the final 1 from A21? Actually A21 yields some numbers ending with 1. We need after that "7 11 6 8 1". Perhaps we can generate them via further expansions after A60 or after A59 chain.

Alright, let's think: after A59 -> A54 A60 we may get sequence: (A54 part) + (A60 part). We may also need to insert a 21 before A60 for the leading 21. So modify rule_1101 (A59 -> A54 A60) to insert 21 between A54 and A60: A59 -> A54 21 A60. Insert 21 (one insertion). But note terminal 21 already appears as a nonterminal? Actually there is a terminal 21 (the integer). So we can insert that.

Now A60's production we can use rule_1445: A43 -> 12 8 9 10 A21 or rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8? Actually rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8. That includes 7 11 6 8 after A21. Perfect! That matches exactly the tail after the final 1 from A21 (which yields 3 5 4 2 1). If we use rule_1104, after 12 8 9 10 A21, we have terminals 7 11 6 8, producing after the A21's final 1: 7 11 6 8. Then final 1 remains? Actually after 8 we need final 1 (the target ends with ... 8 1). The rule does not include final 1. However after the 8 we could have A60-> ... again? Actually the rule ends with 8, but we could then have additional production after A60? Actually A60 after this rule is terminal and not followed by any nonterminals. So the derivation would end after "8". However target ends with "8 1". To get final 1, we could add a terminal "1" after the 8 by inserting in this rule: modify rule_1104 to include a trailing "1". Or we could have A60 -> A60 A58 recursion that eventually yields 1 at the end. But simpler: insert "1" at the end of rule_1104.

So modify rule_1104 (A60 -> 12 8 9 10 A21 7 11 6 8) to "A60 -> 12 8 9 10 A21 7 11 6 8 1". Insert terminal 1 at end (one insertion). This yields final 1.

Now we need to ensure the sequence after A21 yields 3 5 4 2 1 (which matches target). A21 as described can generate 2-5 numbers ending with 1. Use sequence 3 5 4 2 1: produce via (A22->3) A21 -> A22 A21 (then A22->5, etc). So that works.

Now we still need to generate the leading 21 before the block 12 8 9 10. However the target after part "22" includes "21" before 12. Indeed after "22" we need "21 12 8 9 10 ...". So we inserted 21 between A54 and A60 via rule_1101. However A21 will also generate a final 1; but we also have inserted an extra 21 - is there any other 21? Target includes two 21's: one at position 15 (starting after 25?), and later after 22 we have another 21 (position 27?). Let's double-check positions:

Target S after prefix (positions overall 1-40) details we enumerated:

Positions:

1:34
2:31
3:14
4:33
5:29
6:28
7:27
8:26
9:1
10:13
11:32
12:30
13:31
14:25
15:21
16:16
17:24
18:19
19:17
20:18
21:20
22:1
23:15
24:23
25:22
26:21
27:12
28:8
29:9
30:10
31:3
32:5
33:4
34:2
35:1
36:7
37:11
38:6
39:8
40:1

We accounted for 21 at positions 15 and 26.

Our plan: A54's inserted seq yields 25 21 16 24 19 17 18 20 1 15 23 22. That covers positions 14 (25),15 (21),16 (16),17 (24),18 (19),19 (17),20 (18),21 (20),22 (1),23 (15),24 (23),25 (22). That's good.

Then we inserted a terminal 21 before A60 (via rule_1101) to generate position 26 (21). Good.

Now A60's block yields "12 8 9 10 A21 7 11 6 8 1". A21 then yields 3 5 4 2 1. So the sequence after 21 is: 12,8,9,10, then A21 yields 3,5,4,2,1, then 7,11,6,8,1.

So that yields positions: 27=12, 28=8, 29=9, 30=10, then 31-35= 3,5,4,2,1, then 36-39=7,11,6,8, then 40=1. Perfect! That matches target exactly.

Thus overall we have a complete derivation using few inserted terminals:

- In rule_1387 (A66 -> 34 14 33 A25 13 32) we need to insert "31" after 34 to get "34 31 14 33 A25 13 32". That is 1 insertion (terminal 31). 

- In rule_1028 (A45 -> A49 A59) we need to insert "30 31" after A49 to generate "A49 30 31 A59". That's 2 insertions: terminals 30 and 31 (both before A59). However, note that we also need a 30,31 after the prefix but before A59; these terminals not previously present. So count 2.

- In rule_1073 (A54 -> 25 21 24 A23 15 23 22) we need to insert 5 terminals: "16" after 21, and "19 17 18 20" after 24. That's 5 insertions.

- In rule_1101 (A59 -> A54 A60) we need to insert a terminal "21" between A54 and A60. That's 1 insertion.

- In rule_1104 (A60 -> 12 8 9 10 A21 7 11 6 8) we need to insert terminal "1" at end. That's 1 insertion.

Now check other needed adjustments: Also have to insert "31" after 34 in rule_1387 (we already did). But the prefix also needed "30 31" after the A49->... portion (which is after the 32). Indeed after base A49's RHS (producing 34 31 14 33 A25 13 32) we inserted "30 31". That yields after 32 the needed 30,31.

But we need to make sure after A49's expansion we have the numbers "30 31". Indeed our modified rule_1028 gives: A45 -> A49 30 31 A59. So after A49's RHS yields sequence ...13 32, then we insert 30,31, then proceed to A59 which will generate rest.

Thus we also need to ensure A49 productions deliver "34 31 14 33 A25 13 32". This is done by modifying rule_1387 (but note A49's production is not directly referencing A66. We need to connect A49 to A66.

Check: Our derivation used A45 -> A49 A59. A49's expansion must produce the prefix part. However we haven't connected A49 to A66; currently A49 does not directly produce A66; but earlier we considered using A45 -> A49 A59, and we said to modify A49? Actually we intend to use A45 -> A49 A59, and then let A49 have a production that yields the needed prefix; we could modify A49's rule to incorporate A66. For example, rule_1045: A49 -> 34 14 A25 32 30. But we could modify rule_1046: "A49 -> 34 14 33 A25 13 31". But we need "34 31 14 33 ...13 32". So we might change some existing rule for A49 that already contains some needed numbers and insert missing ones.

However, earlier we used A66 for prefix. Hmm let's re-evaluate.

Our current plan used A66 to produce "34 31 14 33 A25 13 32". But we haven't linked A66 to any part of the derivation. How do we incorporate A66? It resides under A3 -> ... via A61 -> A62 -> A66. If we stick with that, we could have A3->A61->... generate prefix via A66. Then use A45 method for suffix? But we currently using A45 as part of A3 derivation, but A3 can't produce both A66 and A45 simultaneously unless we use recursion. Actually we could incorporate A66 via A62, but we may not need to involve A45; perhaps we can have A3 -> A61 (leading to A62->A66) for prefix, then after A66 we'll have recursion to A65 (via A66 -> A66 A65) and A65 can produce A54 (via A65 -> A71 A67) ??? Wait not directly.

Thus perhaps better to unify everything under A45 path: Use A45 -> A49 A59 where A49 will produce the prefix includes the needed numbers (34,31,14,33,...) by inserting necessary terminals into one of its productions, not using A66. That way we avoid using A66 and reduce insertions.

Let's examine A49's existing productions:

- rule_1045: 34 14 A25 32 30
- rule_1046: 34 14 33 A25 13 31
- rule_1047: 34 A25 30
- rule_1044: 1
- rule_1048: A45 A57
- rule_1049: A46 A49
- rule_1050: A49 A48
- rule_1051: A53 A50

Thus A49 can already produce numbers: 34,14,33,13,31,30, etc. It does not have 31 after 34, but we can choose rule_1045 and insert 31 after 34 and before 14, also insert 33 after 14 maybe? But rule_1045 currently has "34 14 A25 32 30". It lacks 33,13,31, and we need 31 after 34, then 14, then 33, then A25, then 13, then 32, then 30,31 (two more). So we could modify rule_1045 significantly: insert 31 after 34, insert 33 after 14, insert 13 before 32 maybe? Actually rule_1045 currently after A25 is 32 then 30. We need after A25: 13 32, then we will insert 30 31 later via rule_1028. So we can modify rule_1045 by inserting "31" after 34 (1 insertion) and "33" after 14 (1 insertion) and "13" after A25 (1 insertion). That would achieve prefix up to A25 13 32, then later we add 30 31 between A49 and A59 as before.

Now rule_1045 would become: A49 -> 34 31 14 33 A25 13 32 30

But we also have a trailing 30 (original), and we will later insert 31 after that via rule_1028. So final sequence from A49 would be: 34,31,14,33, (A25),13,32,30. Good. Then based on rule_1028 insertion we add "31" after A49 (or after its RHS?), but after A49 produced this sequence, we will then have "31" from insertion then move to A59.

But note that the inserted "31" after A49 may conflict with the 30 trailing 30 at end of A49? Since the 30 is part of A49 RHS; after expanding A49 we have ...32 30. Then we insert "31" (via A45 rule insertion) after A49, resulting ...32 30 31 (good!). Perfect.

Now the only remaining is to generate prefix up to A25 (including the 29-28-27-26-1). That's taken care of by A25 -> A26 A25 recursion (no insert needed). Good.

Thus we don't need A66 at all. So we can ignore A66 modifications and use A49 modifications only.

Now count inserted terminals:

- Modify A49 rule (choose e.g., rule_1045) to add 31 after 34, insert 33 after 14, insert 13 after A25. That's 3 insertions.

But we need to also ensure that the A49 rule we modify is the one that is used. We'll modify rule_1045 (A49 -> 34 14 A25 32 30) to become "A49 -> 34 31 14 33 A25 13 32 30". Wait we inserted 31 after 34 (makes "34 31 14 ..."), inserted 33 after 14 (makes "34 31 14 33 ..."), inserted 13 after A25 (makes "...A25 13 32..."). So we also need to keep the original 32. So final RHS: 34 31 14 33 A25 13 32 30.

Thus we inserted 31, 33, 13.

Now we must also adjust A45 -> A49 A59 insertion of "30 31"? Actually we already have 30 in A49 now; we need 31 after that. So we need to insert 31 after A49 before A59. But we also may need to consider that we already inserted the 30 from A49; but we removed original 30 in A49? No original A49 had 30 as terminal, we left it. So after A49's expansion we have ...32 30. Then we need to add 31: so modify rule_1028 to "A45 -> A49 31 A59". Wait we need to only insert "31", not both 30 and 31, because 30 already present. Actually earlier we thought we needed to insert both 30 and 31 after A49, but now 30 is already from A49's original RHS, so only insertion needed is "31". Let's confirm: Original A49 RHS includes a 30 terminal at end. So after A49 expansion, we have trailing 30. Needed sequence after 32: 30 then 31. So we need to insert "31" after the 30, not before. So modify rule_1028 to add a terminal 31 after A49 (i.e., "A45 -> A49 31 A59") i.e., insert 31 after A49. That's 1 insertion.

Thus far inserted: 31 after A49 (one insertion), plus 3 insertions in A49 rule, total 4 insertions.

Now we still need modifications in A54 rule (5 insertions) and insert 21 in rule_1101 (1 insertion) and insert 1 at end of A60 rule (1 insertion). Also modifications to A60 maybe needed to produce the final 1 after 8; we will insert that.

Total inserted terminals: 3 (A49) + 1 (rule_1028) +5 (A54) +1 (rule_1101) +1 (A60) = 11 insertions. Let's check if any other missing terminals remain: Did we need to insert 16 in A54 (we already inserted), 19,17,18,20 (we inserted). Good. Need to insert 21 before A60 (done). Also need to ensure A60 rule includes the extra 1 at end (done). Also need to ensure A45 -> A49 A59 (modified) also yields A59 using proper rule to generate rest.

Now what about the prefix 34 31 14 33 ... does A49 rule also need an insertion of "31" after 34; we did that. But we also need 33 after 14; inserted.

But note we also need to confirm that after 14 we need 33, not 33 after A25? In target it's "34 31 14 33". So our modified A49 yields exactly that: 34,31,14,33,...

Now after 33 we need A25 which yields "29 28 27 26 1". That's okay. Then we need "13" after A25; we inserted 13. Then we need "32". Since after A25 we inserted 13 before 32, we have ...13 32 which matches target position 10 and 11 (13,32). Good. Then we have "30 31" as discussed.

Thus prefix part solved.

Now we need to ensure that A25 yields exactly "29 28 27 26 1". Does A25 generate yes: A25 -> A26 A25 (choose A26 -> 29), then recursively -> 28,27,26, then -> 1. Good.

Now after 30,31 we go to A59.

A59 rule: we need to produce the rest: "25 21 16 24 19 17 18 20 1 15 23 22 21 12 8 9 10 3 5 4 2 1 7 11 6 8 1". That's many numbers.

With our plan: A59 -> A54 A60 (original rule). Insert "21" between A54 and A60 (rule_1101). So the order will be: A54 (gives 25 21 16 24 19 17 18 20 1 15 23 22), then 21, then A60 yields 12 8 9 10 A21 7 11 6 8 1 (since we modified A60 to include trailing 1). That yields final segment exactly.

Now check A54 after modifications: originally "25 21 24 A23 15 23 22". We inserted "16" after 21 and "19 17 18 20" after 24. So we have "25 21 16 24 19 17 18 20 A23 15 23 22". Since A23 -> 1, we will have "25 21 16 24 19 17 18 20 1 15 23 22". Perfect.

Thus A54 yields up to "22". Good.

After A54 we have inserted "21". So sequence continues with 21 (this gives the second 21). Good.

Now A60 as modified: "12 8 9 10 A21 7 11 6 8 1". A21 can produce 3 5 4 2 1. Let's ensure A21 can produce exactly that: It must produce a sequence of A22 and then 1 at the end. So we need to produce 3,5,4,2,1. This can be done using A22 -> 3,5,4,2 (in any order) then A21 -> 1 termination. So we need to select the order: first 3, then A21->A22 A21 again with A22->5, etc. That's possible.

Thus overall sequence will be produced.

Now verify no extra terminals are produced beyond target:

Check each nonterminal expansions:

- A1 -> A2 (no terminals)
- A2 -> A78 A3 (no terminals)
- A78 -> A80 A79 => A80 -> 1, A79 -> 1 . That gives exactly "1 1". Good.

- A3 -> A61? Wait we have not defined path from A3 to A45 etc. Our current plan used A3 to eventually get to A45. We need a derivation: A3 -> A44 -> A45 -> ...? Or A3 -> A61 -> ...? Let's see.

We currently have to produce the suffix using A45. But A3 could be turned into A44 (via rule_68). Then A44 -> A45 (via many rule_768-... mapping). So we can use A3 -> A44 -> A45. Good. So path: A3 -> A44 (rule_68). A44 has many productions A44 -> A45 (rules_768-801 etc). Choose one of them (no terminals) to get A45. So we have no extra terminals.

Now A45 -> A49 A59 (choose rule_1028). After modifying rule_1028 to "A45 -> A49 31 A59", we have after A49, terminal 31 inserted, then A59.

Now we need to ensure that after A45 we also have no extra terminals inserted. Good.

Now A49's updated rule_1045: "A49 -> 34 31 14 33 A25 13 32 30" includes a terminal sequence; great.

Now after A49 expansions, we have the derived sequence as desired: [34,31,14,33, (A25 yields numbers), 13,32,30] then from rule_1028 inserted 31, then A59.

Thus after that we have 31 (the one we inserted) then A59's derivation.

Now A59 -> A54 A60 (choose rule_1101, modified to "A59 -> A54 21 A60"). Insert 21 after A54. So yields A54 expansion then 21 then A60.

A54's modified rule_1073 yields correct numbers up to 22.

Now A60 modified rule_1104 yields "12 8 9 10 A21 7 11 6 8 1". Great.

Now A21 is left to produce the remaining numbers: 3 5 4 2 1. This is all numbers 2-5 plus terminal 1.

Now verify each of these expansions doesn't add any extra terminals besides those we inserted.

Potential extra terminals: some productions we used may have nonterminals we haven't defined fully (like A25 already defined, A21 defined). No other terminals appear.

Thus total insertions = 3 (A49) +1 (A45) +5 (A54) +1 (A59) +1 (A60) = 11.

We should also consider that A44 has many alternative rules; we don't need any insertions there.

Potentially we might also need to insert a terminal (maybe 31) after A45? Already accounted.

Now need to check that rule_1028 modification adds a terminal after A49 but before A59: we said "A45 -> A49 31 A59" (i.e., insert terminal 31). That is consistent with the needed 31 after the 30 from A49. Indeed A49 ends with 30; then we insert 31, then go to A59. So the derived sequence after A49 is "...30 31 ..." correct.

Thus A45's rule no longer includes A49 31 A59; we must note that the original RHS was "A49 A59". Adding terminal "31" yields "A49 31 A59". Good.

Now, check if there is any conflict with A45 also having other productions maybe used? Not needed.

Thus final answer: Updated production rules entries:

- rule_1045 (modified A49)
- rule_1028 (modified A45)
- rule_1073 (modified A54)
- rule_1101 (modified A59)
- rule_1104 (modified A60)

We need to write each rule with its updated RHS, specifying inserted terminals.

Now we need to write new lines for each updated production rule (including rule_<n>:). Should not rewrite unchanged rules.

Thus we produce:

Updated production rules:
rule_1045: A49 -> 34 31 14 33 A25 13 32 30
rule_1028: A45 -> A49 31 A59
rule_1073: A54 -> 25 21 16 24 19 17 18 20 A23 15 23 22
rule_1101: A59 -> A54 21 A60
rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 1

We might double-check rule numbers: The original A60 rule 1104 is "A60 -> 12 8 9 10 A21 7 11 6 8". We inserted "1". So final becomes as above.

Now we need to confirm that modifying rule_1073 exactly as we described: original "25 21 24 A23 15 23 22". Insert "16" after 21 (which is after second terminal) and "19 17 18 20" after 24. So result: "25 21 16 24 19 17 18 20 A23 15 23 22". Good.

Now check: we inserted five terminals: 16, 19, 17, 18, 20. Note order: after 24 we inserted 19 17 18 20; this matches target order 19 17 18 20 after 24. Yes.

Now A49 rule: inserted 31 after 34, inserted 33 after 14, inserted 13 after A25. Wait "A25" appears after 33, we inserted 13 after A25 before 32. The original A49 rule: "34 14 A25 32 30". Insert 31 after 34 (makes 34 31 14 ...). Insert 33 after 14 (makes ...14 33...). Insert 13 after A25 (makes ...A25 13 32...). That's good.

Now check final A49 RHS: "34 31 14 33 A25 13 32 30". Good.

Now we also must ensure there is no conflict: the original also had 30 at the end; we preserve.

Now the derivation yields:

A1 -> A2 -> A78 A3

A78 -> A80 A79: 1 1

A3 -> A44 -> A45 -> A49 31 A59

A49 produces: 34 31 14 33 (A25) 13 32 30

A25 recursively yields 29 28 27 26 1

Thus after A49 we have after 30: then inserted terminal 31

Now A59 -> A54 21 A60

A54 (modified) yields: 25 21 16 24 19 17 18 20 1 15 23 22

Terminal 21 (inserted) yields second 21

A60 yields: 12 8 9 10 A21 7 11 6 8 1

A21 yields 3 5 4 2 1

Thus the full terminal string is:

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

Comparing to target (full sequence) includes "34 31 14 33 29 28 27 26 1 13 32 30 31 ..." yes matches.

Thus we have succeeded.

Now count inserted terminals. 3 + 1 + 5 + 1 + 1 = 11.

Thus final answer updates those rules.

Now ensure we didn't miss any rule number modifications: rule_1028 is the correct rule number for A45 -> A49 A59. Indeed earlier listing: "rule_1028: A45 -> A49 A59". Yes.

Check rule numbers:

- rule_1045: for A49 -> 34 14 A25 32 30 . Yes.

- rule_1073: for A54 -> 25 21 24 A23 15 23 22.

- rule_1101: for A59 -> A54 A60.

- rule_1104: for A60 -> 12 8 9 10 A21 7 11 6 8.

Let's double-check rule number context: Rule_1104 is indeed "A60 -> 12 8 9 10 A21 7 11 6 8". Yes.

Thus we respond with updated lines.

We need to output:

Updated production rules:
rule_1045: A49 -> 34 31 14 33 A25 13 32 30
rule_1028: A45 -> A49 31 A59
rule_1073: A54 -> 25 21 16 24 19 17 18 20 A23 15 23 22
rule_1101: A59 -> A54 21 A60
rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 1

Make sure format matches exactly: "rule_<n>: <RHS>" includes spaces as in original.

Thus answer.Updated production rules:
rule_1045: A49 -> 34 31 14 33 A25 13 32 30
rule_1028: A45 -> A49 31 A59
rule_1073: A54 -> 25 21 16 24 19 17 18 20 A23 15 23 22
rule_1101: A59 -> A54 21 A60
rule_1104: A60 -> 12 8 9 10 A21 7 11 6 8 1