We have a CFG that originally was designed to generate some sequence of integers (from 1 to 19). But some terminal symbols have been removed from the production rules. Our goal is to insert terminal symbols into existing production rules (anywhere) to allow the grammar to produce the given target sequence. We cannot delete or reorder symbols in existing productions, nor can we add new productions. We can only insert terminals into any right-hand side.

Goal: Minimize the total number of inserted terminals across all rules.

We need to find a parse of target sequence starting from A1 using the given rules, possibly with modifications: when a rule's RHS currently consists of non-terminals, we can add terminals between symbols or at ends. That gives more flexibility. We need to determine which missing terminals prevented generation.

First, let's parse the given grammar and target sequence. Let's understand the grammar. The terminals are numbers 1 to 19 inclusive. Currently some terminals appear in rules: see rule definitions with numbers.

The target sequence is:
[7, 8, 6, 1, 18, 1, 19, 10, 5, 7, 2, 6, 9, 16, 15, 4, 11, 7, 8, 6, 3, 17, 1, 13, 1, 12, 1, 14]

Check the count: length = let's count:
Indices:
1:7
2:8
3:6
4:1
5:18
6:1
7:19
8:10
9:5
10:7
11:2
12:6
13:9
14:16
15:15
16:4
17:11
18:7
19:8
20:6
21:3
22:17
23:1
24:13
25:1
26:12
27:1
28:14

Length = 28.

We need to generate this exactly.

First, look at the start rule: rule_1: A1 -> A23 A3 A21 A22 A19 A2 A20

Thus the start expands to those six non-terminals in order:
- A23
- A3
- A21
- A22
- A19
- A2
- A20

Wait that's 7 non-terminals: A23 A3 A21 A22 A19 A2 A20 (total 7). Yes.

Goal: produce target sequence from this expansion. So the concatenation of expansions of each of those non-terminals must equal target sequence.

Thus we need to find expansions for each of these non-terminals.

First get what each currently can generate (given that missing terminals might be inserted). But we need to decide minimal insertions.

Let's examine each non-terminal's production rules:

- A23: rule_59: A23 -> A7 A21 18 A24

Thus A23 expands to A7 A21 18 A24. That's a nonterminal, nonterminal, terminal 18, nonterminal. So A23 yields the concatenation of expansions of A7 then A21 then terminal 18 then A24, plus possible insertions.

- A3: rules:
rule_4: A3 -> 1
rule_5: A3 -> 4 A4
rule_6: A3 -> 7 A5 6
rule_7: A3 -> 10 A6

Thus A3 can produce simple sequences of lengths 1 or 2 with non-terminals.

- A21: rule_56: A21 -> 1; rule_57: A21 -> 16

Thus A21 produces terminal 1 or 16.

- A22: rule_58: A22 -> A7 17

Thus A22 yields A7 17.

- A19: rule_52: A19 -> A7 13; rule_53: A19 -> A11; rule_54: A19 -> A15

Thus A19 can produce A7 13, A11, or A15.

- A2: rule_2: A2 -> A7; rule_3: A2 -> A11
Thus A2 -> either A7 or A11.

- A20: rule_55: A20 -> A7 14

Thus A20 yields A7 14.

Thus the overall start can be:

A23 (-> A7 A21 18 A24)
A3 (-> something)
A21 (-> something)
A22 (-> A7 17)
A19 (choose branch)
A2 (choose)
A20 (-> A7 14)

Thus we need to break down the target sequence into segments for each of those.

Target: 7 8 6 1 18 1 [7 ???]. Let's annotate.

We'll try to see if the start expands as:

Segment1: from A23 = A7 A21 18 A24.

Segment2: from A3.

Segment3: from A21.

Segment4: from A22 = A7 17.

Segment5: from A19 (choose one production maybe A11 or A15 or A7 13).

Segment6: from A2 (A7 or A11).

Segment7: from A20 = A7 14.

Thus total expansions must be concatenated.

Observation: The target includes many patterns like "7 8 6 1 ...". Looks like a typical pattern "7 X 6 1" appears a few times, reminiscent of rule A3 -> 7 A5 6, and other similar patterns.

The target begins with "7 8 6 1 18 1". Let's see: A23 begins: A7 A21 18 A24.

- A7 likely generate something starting with 7 ... . Indeed rule_16: A7 -> 1; rule_17: A7 -> 4 A9; rule_18: A7 -> 7 A8; rule_19: A7 -> 10 A10 9. So the only rule that produces a leading terminal 7 is rule_18: A7 -> 7 A8. So A7 could generate "7" then A8. So we expect A7 possibly yields "7" and then A8 which could produce "8 6 1" perhaps? Indeed A8 has rule_22: A8 -> 8; rule_20: A8 -> 1; rule_21: A8 -> 2 4 A9 3; rule_23: A8 -> 8 10 A10 9. The pattern "8 6 1"? Not directly. But A8 can produce "8"? That's just terminal 8. But our target initially is "7 8 6 1 ...". Could be A7 -> 7 A8 where A8 -> 8 followed by something that yields "6 1"? But there's no direct continuation: In the RHS of A7, after 7 A8 no extra terminal. But A8 expansion would be "8". After that, next part is coming from next nonterminal after A7 within A23: A21. So we might need to insert terminals between A7 and A21 or inside A8 to produce "6 1"? Wait, we need to see how to produce "7 8 6 1" as a concat of expansions of A7 and A21 and maybe some insertions.

A23's expansion: A7 A21 18 A24. So concatenated expansion is:

- A7 expands to something (maybe "7" and something else)
- A21 expands to something ("1" or "16")
- then terminal "18"
- then A24 expansion.

So we need to parse target accordingly.

The target's first part "7 8 6 1 18 1". This could be:

- A7 expands to "7 8 6"? Note A7 -> 7 A8, and A8 could expand to "8 6 1"? But A8 currently cannot produce "8 6 1" (it only has 8, 1, 2 4 A9 3, or 8 10 A10 9). Insertions allowed: we could insert terminals inside A8's RHS. So A8 -> 8 currently. We could insert "6 1" after "8"? Yes, we can insert any terminals anywhere in the RHS. For rule A8 -> 8, we can change it to "8 6 1". That would allow A8 produce exactly "8 6 1". However we need to check also the rest of the sequence: after the "18 1 ...". Actually A23's A21 after A7; A21 expansion is either 1 or 16, but the target after the initial "7 8 6" is "1"? Indeed after "7 8 6" we have "1". So if A7 produced "7 8 6" and A21 produced "1", that matches "7 8 6 1". Then A23's terminal 18 yields "18". Then A24 would need to produce "1"? Let's see the target after "7 8 6 1 18" we have "1". Indeed target: [1]=7, [2]=8, [3]=6, [4]=1, [5]=18, [6]=1, [7]=19... Wait check: after 18 at position5, position6 is "1". So after A23's components (A7, A21, 18, A24), we have consumed up to 6 tokens: 7,8,6,1,18,? maybe the next token (position6) should be from A24. So A24 should generate "1". A24 is defined as: rule_60: A24 -> A7. That's it. So A24 expands to A7. So then A7 should generate "1"? But A7 cannot produce "1" directly? Let's see: A7 rules: A7 -> 1 (rule_16). Yes, A7 has rule_16: A7 -> 1. Good. So A24 -> A7 -> 1 can produce "1". So A24 yields "1". Thus A23 as a whole yields: A7 (7 ? ), A21 (something), 18, then A24 (1). We need to match "7 8 6 1 18 1". So we can assign:

- A7: "7 8 6" (by A7->7 A8, with A8->8 with inserted "6"? Actually how to get "6"? Could be inserted after 8. Then we need "6". Wait but A7 also has only 7 then A8. So A8 must produce "8 6". But then we have also need a following "?" Actually the sequence after "7 8 6" is "1". That is from A21. So A7 should produce "7 8 6". Then A21 produce "1". However we need "6" as part of A7's expansion. So A8 must produce "8 6". That's okay: modify rule_22 (A8->8) to "8 6". Then A7->7 A8 yields "7 8 6". That matches the first three tokens.

But A8->8 6 produces "8 6" (two terminals). However then after that, there is a "1". That belongs to A21. Good.

Now verify A21: must produce "1". It can via rule_56: A21->1. So no modifications needed.

Then we have terminal 18 from A23, matches target token5 (18). Then A24 -> A7 must produce "1". A7->1 via rule_16 is "1". So no modifications needed. Thus the first 6 tokens are accounted for: 7(A7),8(A8),6(A8),1(A21),18,1(A24->A7). Great.

Thus for A23 (through A7 and A8) we just need one insertion: in rule_22 (A8->8) we might insert a "6" after "8". That would be 1 insertion.

But we must also ensure that later parts of the parsing for the rest of the target also align. Let's keep track.

Thus far we used modifications: rule_22: currently "8". Insert "6" after "8" -> "8 6". That's one insertion.

Alternatively could also insert "6" after the whole A8? Actually rule is "A8 -> 8". We can add "6" anywhere, i.e., after 8 or before etc. To produce "8 6". Yes.

Now, we need to parse the rest.

Our segmentation:

Target positions:
1:7 (A7 part)
2:8 (A8 part)
3:6 (A8 part)
4:1 (A21)
5:18 (A23)
6:1 (A24 via A7)

Thus after A23 we have consumed positions 1-6.

Next is A3 (rule_1's second non-terminal). A3 expansions:

- rule_4: A3 -> 1
- rule_5: A3 -> 4 A4
- rule_6: A3 -> 7 A5 6
- rule_7: A3 -> 10 A6

Thus A3 can produce a sequence maybe like "7 A5 6", or "4 A4", or "1", or "10 A6". The next target token after position6 is "19". Let's verify: target at position7 is 19. So we need A3 to produce starting with 19, but none of the current productions produce 19 at start. The only terminal numbers currently in production rules that are 19? Actually 19 is not present anywhere in the rules. The goal maybe to insert "19" somewhere. Since we can insert terminals in any RHS, we could insert terminal 19 into A3's production's RHS. Which rule to modify to accommodate 19? Also note that after A3, we have A21 (the third non-terminal in rule_1), then A22, etc. So maybe A3 should generate exactly "19" (i.e., a single terminal 19), then A21 yields something else. Or A3 could generate "19" followed by something else. Or A3's production could produce "19" as part of its RHS.

Let's examine target sequence after pos6: we have "19 10 5 7 2 6 9 16 15 4 11 7 8 6 3 17 1 13 1 12 1 14". So the remainder subsumes many tokens.

The total sequence for A3 + A21 + A22 + A19 + A2 + A20 must equal all these tokens.

Thus we need to parse these as:

A3: to produce perhaps part of them.

A21: after A3.

A22: after that.

A19: after that.

A2: after that.

A20: after that.

So we need to allocate lengths.

Let's compute potential expansions:

- A3 can produce several options. Let's estimate possible lengths.

If A3->1 yields length1 (terminal 1). If A3->4 A4 yields at least length1 (if A4->1) -> "4 1". Or if A4->... more. If A3->7 A5 6 yields maybe length three or more depending on A5 expansions. If A3->10 A6 yields at least 2 (10 plus expansion of A6). So many possibilities.

But target starts with "19". So we need to either modify A3 to have 19 as a terminal, either insert before everything (prefix), after everything (suffix) or inside. For minimal insertions likely we use a single insertion: add "19" before the RHS or after. However we must ensure the overall sequence matches the target, so if A3 produces "19 X" and then A21 adds more tokens, we need that to fit exactly.

Alternatively, perhaps "19" could be generated by A21? A21 can produce "1" or "16". No 19.

Or maybe A19? A19 can have production A7 13 which might produce "7 ... 13", not 19. It also can be A11 or A15.

Thus "19" likely must be added somewhere else. Maybe it's part of A2? A2 expands to A7 or A11. A7 has rule that yields "1" or "4 A9" or "7 A8" or "10 A10 9". Not 19.

Thus best is to insert terminal 19 into A3's RHS, or maybe into A21? But we see A21 only has 1 or 16. Can't easily insert a terminal before those? Actually we can insert terminal(s) anywhere, even before the existing terminal. So we could insert "19" before A21's terminal 1 or 16. But placement must be such that when parsing, we still follow order A3 A21 A22... So maybe we could add "19" to A3's production with minimal insert.

Given we have to have "19" appear exactly after everything leading up to that point (i.e., after the first 6 tokens; this is position7). So the boundary between A3 and A21 and A22 etc is after consuming tokens part of A3. That is, if we make A3 produce tokens up to and including "19", then A21 will start with token10 (i.e., next token after "19").

But according to target, after "19", token8 is "10". So A3 could produce "19 10"? Or A3 could produce just "19", and A21 produce "10"? But A21 can't produce "10". So perhaps A3 must produce "19 10". Let's examine possibilities.

A3's production options: we can choose rule_7: A3 -> 10 A6. That starts with 10. It also includes terminal 10 before A6. If we insert "19" before 10, we get "19 10 A6". That would produce "19 10 ..." which matches target: "19 10 5 ...". Good.

Or we could use rule_6: A3->7 A5 6, but we would need to insert 19 into that, but that seems less matched.

Thus a minimal approach: modify rule_7 A3->10 A6 by inserting a "19" before "10". So the RHS becomes "19 10 A6". That's one insertion.

Now A3 yields "19 10" plus whatever A6 yields.

Looking at target after "19 10": we have "5 7 2 6 9 16 15 ..." Actually check target: after "19 10" the sequence continues "5 7 2 6 ...". Let's list target after first 6 tokens:

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

Thus after "19 10", the next token is 5.

Now after A3 (including its A6 expansion), we have A21 (which is the third non-terminal after A3). So A21 must produce some portion of this sequence after the tokens produced by A3. A21 can produce "1" or "16". So it might produce "16" later in the target (position14 is 16). So maybe A21 should produce token16 (which is after a bunch of other tokens). But A21 appears before A22 in rule, so A21's output must appear before that of A22. However in the target, the token "16" appears at position14 which is after many tokens (5,7,2,6,9). Perhaps A21 occurs later? Wait check the order: rule_1 expansions: A23 (positions1-6) then A3 (positions7 onward), then A21 (after A3), then A22, etc. So after A3's tokens we then have A21 tokens, before A22. So if we want A21 to produce token "16", then A21 must be placed in the target location before A22 tokens. But we see that "16" occurs at position14, which is after the tokens from A3 may be longer than just early portion. Let's examine possible segmentation.

Let’s denote length of A3 expansion as L3, length of A21 as L21, length of A22 as L22, length of A19 as L19, length of A2 as L2, length of A20 as L20.

We know total remaining tokens from pos7 to 28 inclusive: count = 28-6 = 22 tokens.

Thus L3 + L21 + L22 + L19 + L2 + L20 = 22.

Let's analyze the possible lengths of each.

- A21: either 1 (terminal 1) or 1 (terminal 16). So length L21 = 1.

- A22: A22->A7 17. A7 can produce at least length1 (terminal 1) maybe longer if we modify. So L22 >=2. But can we produce longer expansions via A7 expansion? A7 expands to either 1 (len1), 4 A9 (len?), 7 A8 (len?), 10 A10 9 (len?). So length can be varied. But at minimum to get 2 tokens, we need at least A7 produce 1 token (1 possible) then terminal 17. So L22 = 2 minimum.

- A20: A20-> A7 14. So at minimum 2 tokens (if A7 yields 1). So L20 >=2.

- A2: either A7 or A11. So minimal length: if A7 yields 1 => L2 >=1 if A2->A7 (but A2->A7 is not exactly rule_2: A2 -> A7; but A7 yields at least 1). So minimum L2 = 1 (via A7->1). Or A2->A11 (A11 may yield 1) so also min 1.

- A3's length: variable based on rule choice.

Now we have constraints: sum L3 + L21 + L22 + L19 + L2 + L20 = 22.

Let's assume minimal other components: L21=1, L22=2, L20=2, L2=1. Then we have L3 + L19 + (1+2+2+1) = L3 + L19 +6 = 22 => L3 + L19 = 16.

A19 can be A7 13, A11, or A15. Let's inspect lengths for each.

- A19 -> A7 13: length = len(A7) + 1. Minimum is A7->1 => length=2. Could be longer with A7->... expansions.

- A19 -> A11: A11 minimal length = 1 (via A11->1). So L19 = 1.

- A19 -> A15: A15 minimal length = 1 (via A15->1). So L19 = 1.

So minimal L19 = 1. But can be longer.

Thus L3 + L19 = 16, so L3 must be ~16 (if L19 minimal). That's a relatively large chunk for A3.

Alternatively, maybe we can allocate some length to L22 or L20 via longer expansions of their A7's, or L2 via longer expansions, or L19 longer, which reduces L3 length.

Our goal is to minimize insertions, not length of expansions. Insertions are needed for specific missing terminals like 19, maybe also others like 6 1 etc.

Now we need to walk through the target to see if we can assign expansions to each non-terminal with minimal changes.

Instead of an abstract approach, we can attempt to find a parse tree by reasoning top-down.

Let’s start with A3 -> 19 10 A6 (modified rule). So A3 yields "19 10" + expansion of A6.

Now look at target after "19 10". The target remaining tokens at pos9 onward are: [5,7,2,6,9,16,15,4,11,7,8,6,3,17,1,13,1,12,1,14] (20 tokens). So A6's expansion must match a prefix of this list, possibly up to a point. After A6's expansion, we will have A21 (which yields 1 or 16). So A6 expansion will produce some tokens right before A21's token.

Now A6 expansions:

- rule_14: A6 -> 1
- rule_15: A6 -> 5 7 A5 6

Thus A6 can be "1" or "5 7 <expansion of A5> 6". A5 expansions:

- rule_10: A5 -> 1
- rule_11: A5 -> 2
- rule_12: A5 -> 2 4 A4 3
- rule_13: A5 -> 8 10 A6

Thus A5 can directly produce "1" or "2", or more complex.

Given target after "19 10" begins with "5". That's promising because A6 -> 5 7 A5 6 produces "5 7 ... 6". Indeed target has "5 7 2 6". Let's check: after 19 10, we have "5 7 2 6". Yes! So it matches A6->5 7 A5 6 where A5 must produce "2". That matches rule_11: A5 -> 2. Perfect, no insertion needed there.

Thus A6 yields "5 7 2 6". Good.

Now after that, the target at position13 is "9". So after A6, we have consumed tokens 7-12? Actually list:

Positions:

7:19
8:10
9:5
10:7
11:2
12:6
13:9
14:16
15:15
...

Thus after A6's "5 7 2 6", we are at position13, token 9.

Now next non-terminal in A1's RHS after A3 (i.e., after A6's expansion) is A21. So A21's expansion must start at position13. A21 can be "1" or "16". The token at position13 is 9, not 1 or 16. So this doesn't match. This indicates that our assignment of expansions where A6 consumes up to token12 is fine, but A21 cannot generate 9. Something is wrong. Perhaps A6's expansion should be longer to include "9" as part of A6? But A6's production currently is 5 7 A5 6. It doesn't produce "9". However we could add an insertion of "9" after that? We are allowed to insert terminals anywhere in the RHS of A6. So we could add "9" after the 6, or before 5, etc. But we need to align where token9 appears. Actually we have "9" after "6". At target sequence after "5 7 2 6" comes "9". This could be produced by inserting "9" after the 6 in the RHS of A6. So modify rule_15: A6 -> 5 7 A5 6 -> add "9" after "6" (or before) to get "5 7 A5 6 9". Or we could modify A6->5 7 A5 6 to be "...6 9". That's one insertion.

Alternatively, we could choose A6->1 (rule_14) and then A21 produce "5"? Not likely.

So we insert "9" after the terminal 6 in rule_15. That's one insertion.

Thus after this insertion, A6 yields "5 7 2 6 9". That matches target positions 9-13: 5,7,2,6,9. Good.

Now after A6 expansion, the next token at position14 is "16". This is for A21. Good: A21 can produce "16" via rule_57. So we can set A21 -> 16 (no insertion needed). So A21 yields token "16". Great.

Thus A21 matches token14.

Now after A21 comes A22, which is A7 17 (rule_58). So A22 yields expansion of A7 then terminal 17. Our remaining target tokens after 16 are at position15 onward:

Token list after position14 (16):
15: 15
16: 4
17: 11
18: 7
19: 8
20: 6
21: 3
22: 17
23: 1
24: 13
25: 1
26: 12
27: 1
28: 14

Thus we need A22 to expand to some prefix of this: A7 must generate some tokens before token 17, and after A7 we have terminal 17 which aligns with token "17" at position22. So we need A7 to generate tokens from positions 15 up to 21 (inclusive) (i.e., tokens 15-21: 15,4,11,7,8,6,3). Wait check: after 16, we have 15,4,11,7,8,6,3, then 17. So the A7 should generate "15 4 11 7 8 6 3". Also we need to get to token 17 exactly.

Check the target from pos15 to pos21: (15)15, (16)4, (17)11, (18)7, (19)8, (20)6, (21)3, then (22)17.

Thus we need A7 to produce "15 4 11 7 8 6 3". Is there a rule that can produce such with minimal insertions? Let's examine A7's productions:

- rule_16: A7 -> 1
- rule_17: A7 -> 4 A9
- rule_18: A7 -> 7 A8
- rule_19: A7 -> 10 A10 9

Thus A7 tends to start with a terminal 1,4,7,10, then sub-non terms.

Our target A7 must start with "15". None of these productions start with 15. So we need to insert terminal 15 somewhere. We can insert "15" before the existing terminal. For rule_16 (A7->1), we could insert "15" before "1" to yield "15 1". But we need "15 4 11 7 8 6 3". That seems we need multiple tokens after 15. Starting with 4 is plausible via rule_17: A7->4 A9. So maybe we can use rule_17 (4 A9) after inserting "15" before it. So we modify rule_17: A7 -> 15 4 A9 (insert 15 before 4). Then A7 yields "15 4 <expansion of A9>". After that we need A9 to generate "11 7 8 6 3". Let's see A9's productions:

- rule_24: A9 -> 1
- rule_25: A9 -> 11 A8 6

So A9 can be "11 A8 6". Perfect for starting with "11". So we can set A9 -> 11 A8 6, which yields "11 <expansion of A8> 6". After that we need tokens "7 8 6 3". So A8 must generate "7 8 6 3"? Wait let's examine: after "11", we have tokens (positions after 11): 7,8,6,3. Actually after "11" (pos17), the next tokens are: pos18=7; pos19=8; pos20=6; pos21=3. So we need A8 to generate "7 8 6 3".

But A8 currently has productions:
- rule_20: A8 -> 1
- rule_21: A8 -> 2 4 A9 3
- rule_22: A8 -> 8
- rule_23: A8 -> 8 10 A10 9

None starts with 7. However we may insert terminals into the RHS. For instance, rule_22: A8 -> 8. We could insert "7" before "8", and also "6 3" after "8"? Actually we need a sequence "7 8 6 3". So we can modify A8's rule_22: "A8 -> 7 8 6 3". That's insertion of "7" before and "6 3" after. That's three insertions (if we start with "8" as base, we add "7" and "6" and "3"). However we could also use rule_21: A8 -> 2 4 A9 3, and insert necessary terminals around. That rule yields "2 4 A9 3". Not start with 7. Harder.

Alternatively, maybe we could use rule_20: A8 -> 1, and insert "7 8 6"? Not ideal.

Better: Use rule_22 (simple "8") and add "7" before, "6 3" after. That would be 3 insertions.

But maybe we can reduce insertions using rule_23: "8 10 A10 9". That's "8 10 ... 9". Not helpful.

Thus likely we need at least three insertions for A8 to produce "7 8 6 3". However perhaps we could produce "7" by using A8's ability to include a nonterminal that later yields "8"? Actually A7 also has rule_18: A7 -> 7 A8. Maybe we could restructure? Wait we are inside A7->15 4 A9. But maybe we could choose a different A7 production. Let's explore alternatives.

We need A7 to produce "15 4 11 7 8 6 3". Could we use A7 -> 7 A8? That would start with 7, not 15. We could insert "15 4 11" before 7? Possibly rule_18: A7->7 A8. Insert "15 4 11" before "7". That yields "15 4 11 7 A8". Then A8 must generate "8 6 3". That seems plausible: if we modify rule_18 (A7->7 A8) by inserting terminals before the 7. Insert "15 4 11". That's three insertions before 7. Then A7 expansion becomes "15 4 11 7 A8". Next A8 must expand to "8 6 3". That's shorter than previous (we only need "8 6 3" instead of "7 8 6 3"). A8 can produce "8" (rule_22) and we need to insert "6 3"? Actually "8 6 3" after "8". That's two insertions: 6 and 3 after 8.

Thus total insertions = 3 (in A7 rule) + 2 (in A8 rule) = 5.

Alternatively, we could use A7->10 A10 9 etc. Not helpful.

Now alternative: Use rule_17: A7->4 A9. Insert "15" before 4 (1 insertion). Then A7 yields "15 4 A9". It then uses A9->11 A8 6. So after inserting "15", we have tokens: 15 4 11 A8 6. Then we need to produce "7 8 6 3". Actually after "6" we need "7 8 6 3"? Wait sequence: 15 (pos15) then 4 (pos16) then 11 (pos17) then 7 (pos18) then 8 (pos19) then 6 (pos20) then 3 (pos21) then token22=17. The A7 currently yields "15 4 11 A8 6". That is: after "11", we have A8 then "6". So tokens: 15 4 11 <A8 expansion> 6.

If we set A8 to expand to "7 8 6 3"? But note there is also an extra "6" after A8 produced by rule. The final tokens after A8 should be a "6". So overall we would have: 15 4 11 <A8 tokens> 6 (terminal at end of A9). The sequence after A8 must be "6"? Actually the target after the "7 8 6 3" is "17". No extra "6". So the "6" from A9 (the terminal after A8) must either be part of the target "6" token at position20. But we already need "6" at token20 (the third token after 8). However we have two "6"s maybe? Let's map.

If we produce: 15 4 11 (from A7 and A9's prefix). Then A8 expands to produce "7 8". That's two tokens. Then the trailing "6" from A9 yields "6". So thus far we have: 15 4 11 7 8 6. That's target positions 15-20: 15 4 11 7 8 6. Good! Then after that we have token21 = 3, and then token22 = 17. So we need a "3" after the trailing 6 before 17. In this derivation, after terminal 6 from A9 we are done with A7 expansion; next comes A22's terminal 17. No "3". So we need to generate "3" between the A9's trailing 6 and the 17. Where could we insert it? We can insert "3" after the trailing 6 in the RHS of A9 -> 11 A8 6. Since rule_25: A9 -> 11 A8 6. We could insert "3" before the final 6, or after it, but that would be after A9's expansion but before the 17 terminal of A22. However the 3 must appear before 17. So we could insert "3" after the 6 in rule_25 (i.e., change RHS to "11 A8 6 3"). Then A9 would yield "11 <A8> 6 3". That would produce the needed "3" after the 6. Then A22's terminal 17 would be next. Good. However need to ensure that the A8 expansion yields the needed "7 8". But A8 currently is "8". We need it to produce "7 8"? Wait we need after "11" a "7 8". The target after 11 is "7 8". So if we set A8 to produce "7 8"? That is doable by modifying rule_22 (A8->8) to "7 8". That's one insertion of "7" before 8. Actually no, we need "7" before 8. So modify rule_22: "8" -> "7 8". Insert "7". That yields "7 8". Then A9-> "11 (7 8) 6 3". Wait we also need the trailing "6" after "7 8"? Yes, that is after A8's expansion, the rule's "6" yields a "6". So we get "11 7 8 6". Then we inserted "3" after 6, giving "11 7 8 6 3". Good. So far we produce "15 4 11 7 8 6 3". That's the exact token sequence we want. Then after finishing A7 (with inserted "15" before 4), after A7's entire expansion we get those tokens, then A22's terminal "17" (already matched token22). So this looks viable.

Now count insertions needed:

- Modify A7 rule_17: "A7 -> 4 A9" become "A7 -> 15 4 A9". That's one insertion of terminal 15 before 4.
- Modify A9 rule_25: "A9 -> 11 A8 6" become "A9 -> 11 A8 6 3". That's one insertion of terminal 3 after the 6.
- Modify A8 rule_22: "A8 -> 8" become "A8 -> 7 8". That's one insertion of terminal 7 before 8.

Thus three insertions. However we also need to ensure that we didn't need any extra insertion for "6"? Actually we have the trailing 6 from A9 rule; it's part of original rule.

Thus total insertions for A22's expansion: 3.

We also earlier inserted "6" into A6 rule_15 to match token 9. So one insertion there.

We inserted "6" (as part of rule_22 initially for A8 at start). Actually we earlier inserted "6" after 8 in A8 (rule_22: "A8 -> 8 6") to get the first "8 6". That's another insertion.

And we inserted "19" before "10" in A3 rule_7 => 1 insertion.

So far total insertions: A8->8 (first) insertion: +1, A6 rule_15: +1, A3 rule_7: +1, A7 rule_17: +1, A9 rule_25: +1, A8 rule_22 (second usage): +1 = total 6 insertions.

Now we need to verify that the rest of the sequence after token22 = 17 matches the remaining tokens: tokens after 17 (pos22) are:

pos23 = 1
pos24 = 13
pos25 = 1
pos26 = 12
pos27 = 1
pos28 = 14

Now after A22 (which gave terminal 17), we have next non-terminal: A19.

Recall A19 options:

- A19 -> A7 13
- A19 -> A11
- A19 -> A15

We need to match the remaining sequence: 1,13,1,12,1,14.

Let’s examine each option:

Option 1: A19 -> A7 13. A7 expands to something then terminal 13, which matches maybe "1 13"? Indeed behind token sequence: "1 13", looks like 1 then 13. So could use A7 to generate "1". Then the terminal 13 part matches token 13. Then the rest (1,12,1,14) would be from A2 and A20.

Option 2: A19 -> A11. A11's productions:

- rule_28: A11 -> 1
- rule_29: A11 -> 4 A12 3
- rule_30: A11 -> 7 A13 6
- rule_31: A11 -> 10 A14 9

Thus to generate "1 13 ..." we could use A11->1 to get a "1". But then we still need "13..." etc. A19 producing only "1". Then after A19 we have A2, then A20. Not a perfect match.

Option 3: A19 -> A15. A15 productions:

- rule_39: A15 -> 1
- rule_40: A15 -> 4 A16 3
- rule_41: A15 -> 7 A17 6
- rule_42: A15 -> 10 A18 9

Similar.

Thus likely best is A19 -> A7 13 (since we have 1 then 13). Let's see if A7 can produce "1". A7 has rule_16: A7 -> 1. So yes, using rule_16 produces "1". So A19 with production rule_52: A19 -> A7 13 will give "1 13". That matches tokens pos23=1, pos24=13.

Thus we can use that. That doesn't need modifications. Good.

Now after that, we have A2 (next non-terminal) and then A20. Let's check remainder after 13. After tokens 1 13, we have "1 12 1 14". That is pos25=1, pos26=12, pos27=1, pos28=14.

So after A19 we have A2: rule_2: A2->A7, rule_3: A2->A11.

We need to generate something that yields "1 12 1 14"? Wait the remainder includes two more non-terminals: A2 then A20 (which yields A7 14). So final A20 will produce an A7 then terminal 14. So the final token 14 matches A20's terminal 14 after A7's expansion. So the token before 14 (pos27) must be from the A7 inside A20. And the token before that (pos26) is 12, which must be from A2 expansion.

Thus the structure: after A19's "1 13", we have A2 (must produce token(s) ending before the A7 of A20). Then A20's A7 produces a token (maybe "1") and then 14 final.

Thus the token sequence after pos24=13 must be: (some token(s) from A2) then (A7's token) then 14.

The target suffix: 1,12,1,14. So perhaps A2 produces "1 12", and A20's A7 produces "1", then 14, gives "1 12 1 14". That seems plausible.

Now, does A2 have a production that can yield "1 12"? A2->A7 (rule_2) or A2->A11 (rule_3). A7 can produce "1"? Yes rule_16: A7->1. But to get "1 12", we need A2 produce two tokens: maybe A2->A7 where A7 produces "1 12"? But A7 expansions currently produce only a single terminal (unless we add insert). Could we modify A7 to produce "1 12"? Yes we can insert "12" after the "1" in rule_16: modify A7->1 to become "1 12". However that would cause A7's expansion to produce "1 12". Then A2->A7 yields "1 12". Then A20's A7 (but which rule for A7?) might produce "1" (without insertion). Then final "14". That matches. That would require one insertion: add "12" after the 1 in A7 rule_16.

Alternatively, we could have A2->A11 and A11 produce "1 12"? Does A11 have productions that could generate 12? Not directly. A11's productions currently produce "1", or "4 A12 3", "7 A13 6", "10 A14 9". None have 12. So easiest is A2->A7 and modify A7->1 to "1 12". That is 1 insertion.

Now also need A20's A7 to be "1". That's fine with rule_16 unchanged (produces "1").

Thus we need add a new insertion to rule_16: A7 -> 1 12 (if we want A7 to produce "1 12").

But hold on: we also used A7 rule_17 and rule_18 and rule_16 earlier. Let's check consistency: A7 is used in multiple places: A23's A7, A22's A7, A19's A7, and A24 and A20. Some of these A7's may be using different rules; we can choose which rule each instance uses based on nonterminal expansions. Since nonterminal A7 has multiple production rules, each occurrence can choose any rule. So modifications to a rule affect all uses of that rule. However if we modify A7->1 by inserting "12", that will affect any usage of that rule (i.e., when we choose rule_16). But we may not use rule_16 for all A7 occurrences: we may use different rules for different occurrences. That is fine. So we can add insertion in rule_16 and still use rule_17 (modified with 15 insertion) for the earlier uses.

Thus we need to insert "12" into rule_16 to produce "1 12". That adds a "12" terminal in A7->1. But note we also have token "12" in the target at position 26. Good.

Thus we will have A2->A7 (using rule_16) yields "1 12". Good.

Now A20's rule_55: A20 -> A7 14. We need to choose A7 production that yields "1". Use rule_16? But we have modified rule_16 to "1 12". That would yield "1 12". But we need just "1". There is also rule_17: A7->4 A9 (modified to "15 4 A9") but that yields longer. rule_18: A7->7 A8 (modified with preceding 15 4 11 maybe? Actually we modified rule_17; we haven't touched rule_18. Rule_18 is "7 A8". That would yield "7 ..." not "1". So we could keep a separate production rule that yields exactly "1" without additional terminals, but we have changed rule_16 to add an extra "12". That disrupts the ability to get plain "1". But maybe we could avoid modifying rule_16 and instead insert "12" elsewhere in A2's expansion, e.g., A2->A7 and then A7 yields "1", then we insert "12" after the A7 in the A2 rule (i.e., modify rule_2: A2 -> A7 to become A2 -> A7 12). Since we have ability to insert terminals into any RHS, we could add the "12" after A7 in rule_2. That would yield "1 12" as needed, while leaving A7->1 unchanged. That is probably better: one insertion into rule_2: A2 -> A7 12. So modify rule_2: add "12" after A7. That yields A2 => A7 12. A7 using rule_16 gives "1". Thus A2 yields "1 12". Perfect. This is one insertion.

Thus keep rule_16 unchanged (A7->1). Good.

Now we need to verify later expansions: A20->A7 14 should produce "1 14". Since A7 yields "1". Good.

Thus final tokens: after A19 (producing "1 13"), A2 yields "1 12", A20 yields "1 14". Sequence: 1 13 1 12 1 14. That matches target suffix.

Thus we don't need to modify rule_16.

Now check all modifications:

- Insert "6" after "8" in rule_22 (first A8) to produce "8 6". So rule_22 becomes "A8 -> 8 6". This insertion yields the first part after "7". Wait careful: The first A8 occurs within A7->7 A8 for A23. Let's re-evaluate: In A23 we had A7 (first) A21 18 A24. For A23 we used A7->7 A8. That A8 we inserted "6" after 8 to give "8 6". That gave A7->7 A8 => "7 8 6". Confirm that matches tokens positions 1,2,3. Indeed very good.

Now the later A8 inside A22's A7->4 A9 -> 11 A8 6 etc. That uses the same nonterminal A8. We modify rule_22 globally (A8 production). However that would affect also the earlier A8 within A23 (since it's the same rule). That's okay because rule_22 will now be "A8 -> 8 6". But we also want A8 to later be able to produce "7 8"? Actually we also plan to modify rule_22 to become "7 8"? Wait earlier we said we need A8 to produce "7 8" for the part in A22. But since rule_22 is a single production, we cannot have two different RHS for same rule; we can only modify it but can't create separate version. However A8 also has other productions: rule_20, rule_21, rule_22, rule_23. The rule_22 is the one we are modifying. But we also need A8 to have two different expansions in different contexts: once for "8 6" (first use), and once for "7 8". But each use of A8 can choose any of its productions: rule_20, rule_21, rule_22, rule_23. So we could choose a different production for the later use, like rule_20 or rule_23, etc. We haven't considered modifications to other A8 productions.

The later A8 we need to produce "7 8". Could we choose rule_20: A8 -> 1, insert "7 8"? That would be "7 8 1"? Not good.

Alternatively, we could use rule_23: A8 -> 8 10 A10 9. Insert "7" before "8"? That yields "7 8 10 A10 9". Too many tokens.

Better to modify rule_22 for the later use: we need it to produce "7 8". But that conflicts with first use (need "8 6").

But maybe we don't need the first use to be "8 6". Let's re-evaluate the first use: we used A7->7 A8 for A23. Instead we could use a different production for that A8, not rule_22. For instance, we could use rule_21: A8 -> 2 4 A9 3, adjusting with insertions to produce "8 6"? That's more complex. Or we could use rule_20: A8 -> 1, but insert "8 6"? Could modify rule_20 to "8 6 1"? That yields extra "1". Not needed.

Alternatively, we might treat the first A8 as using rule_22 and the later A8 also using rule_22, but we need it to produce different strings. That would generate inconsistencies unless we insert extra terminals differently each use? Since the production rule is same for all uses; we can't have it produce two different strings without insertions that vary per occurrence? The modifications to rule apply globally, but we can insert terminals anywhere in the rule; they are fixed, not dependent on context. So if we modify rule_22 to e.g., "7 8 6" (i.e., 7,8,6) then for the first use we would get "7 8 6"? Actually we want "8 6" (or "8 6"), not "7 8 6". The first use needed "8 6". The later use needed "7 8". So mismatches.

Thus we cannot rely on same rule for both uses unless we rearrange to use different productions for each usage. Maybe we can use different A8 production for one of the occurrences. Let's check possibilities.

First A8 (in A23's A7->7 A8). We have A7->7 A8. We need A8 produce "8 6". That could be obtained by using rule_22: A8->8 6 (modify rule_22). Good.

Second A8, which appears within A9->11 A8 6. This is clause for A22's A7->4 A9. We need A9's A8 to produce "7 8". But we could choose a different production for A8: maybe rule_20 or rule_23 or rule_21, with insertions to produce "7 8". Let's examine these.

- rule_20: A8 -> 1. Could insert "7 8" before the "1"? That yields "7 8 1". That gives extra "1". Not good.

- rule_21: A8 -> 2 4 A9 3. Could insert "7" before 2? Then we get "7 2 4 A9 3". That's not "7 8". Not good.

- rule_23: A8 -> 8 10 A10 9. Could prepend "7"? yields "7 8 10 A10 9". That's "7 8 ..." but extra tokens.

Perhaps we can still use rule_22 for the second use but modify it to "7 8"? That would break first use. Unless we use a different A8 production for the first occurrence (i.e., for A23's A8) we choose a different rule, maybe rule_20 with insertions to get "8 6". Let's examine possibilities:

Goal for A23's A8: produce "8 6". Could use rule_20: A8 -> 1, and insert "8 6" before or after to get "8 6 1"? That would add extra "1". Not ideal. Could we insert "8 6" after 1? yields "1 8 6". That would give "7 1 8 6"? But we need "7 8 6". That has "1" undesirable.

Alternatively, rule_23: A8->8 10 A10 9. Insert "6" after "8"? Then we have "8 6 10 A10 9". That yields extra "10 A10 9". Not good.

But maybe we could use rule_21: "2 4 A9 3"? Insert "8 6"? Not good.

Thus best for A23's A8 is rule_22.

But we also want A9's A8 to produce "7 8". Perhaps we can modify rule_22 globally to produce "7 8 6"? That would give the first A8 "7 8 6". But we need "7 8 6"? Wait first A8 occurs after "7". A7->7 A8: If rule_22 becomes "7 8 6", then A7->7 A8 yields "7 7 8 6". That's "7 7 8 6" (two 7s). Not match.

Alternatively, we could not modify rule_22 but rather modify rule_25 in A9 to generate needed "7 8". Actually A9->11 A8 6 3. We need A9 produce "11 7 8 6 3". Using rule_22 for A8 yields "8 6". That produces "11 8 6 6 3"? Actually with rule_22 as "8 6", we get "11"+"8 6"+"6"+"3" = "11 8 6 6 3". Not match.

Thus we need A8 inside A9 to produce "7 8". So we need a different production for that A8 occurrence, perhaps rule_20->1 with insert "7 8"? That yields "7 8 1". But then the A9 expansion would be "11 7 8 1 6 3"? Actually A9->11 A8 6 3, with A8->7 8 1 gives "11 7 8 1 6 3". That's "11 7 8 1 6 3". Not match target "11 7 8 6 3".

We need "11 7 8 6 3". Following A9 rule: after A8, we have a 6 terminal. So if A8 yields "7 8", then we have "6" from A9. So "11 7 8 6 3". That's correct. So we want A8 to yield "7 8". So we need a rule for A8 that can produce "7 8". We could create a new production by inserting terminals into any existing rule. Since we cannot add new rules, we must modify an existing rule to yield "7 8". Perhaps we can modify rule_20 (A8->1) to "7 8". But then A8 would produce just "7 8". However rule_20 currently "1". Inserting "7" before "1" and "8" after maybe? We could replace "1" with "7 8"? Actually we can't delete "1". Only allowed to insert terminals. So rule_20->1 becomes "7 1 8"? Actually we can insert terminals before or after, but we cannot remove "1". So cannot change it to just "7 8". Unless we insert "7" before and "8" after "1": yields "7 1 8". That doesn't match.

Alternatively, modify rule_23: "A8 -> 8 10 A10 9". Insert "7" before leading "8"? yields "7 8 10 A10 9". Not good.

Modify rule_21: "2 4 A9 3". Insert "7 8" somewhere? Not feasible.

Thus likely we need to use rule_22 to produce "7 8". Let's modify rule_22 globally to "7 8". But we need also the first A8 to produce "8 6". That's not possible simultaneously.

Thus we need two distinct A8 expansions; but we have only four production rules for A8: rule_20 (1), rule_21 (2 4 A9 3), rule_22 (8), rule_23 (8 10 A10 9). We can choose which rule to use at each place. However we need to achieve both "8 6" and "7 8". None of these productions currently produce those strings. However we can insert terminals to any RHS. Perhaps we can modify rule_20 to "8 6"? Insert "8 6" before or after the 1, but can't delete 1. So it would produce something like "8 6 1" or "1 8 6". That's 3 tokens, not ideal.

Alternatively, modify rule_22 to be "8 6" for the first A8, and then modify rule_21 to produce "7 8"? Let's examine rule_21: A8 -> 2 4 A9 3. Could insert "7" before "2"? That yields "7 2 4 A9 3". Not "7 8". Could we use the nonterminal A9 to derive "8"? Possibly. A9 -> 1, or A9 -> 11 A8 6. A9 could be 1. That wouldn't produce 8. Or could be 11 A8 6 (which involves A8 again...). That would cause recursion. But perhaps we can produce "7" from insertion and "8" from A9's expansion? This gets complicated.

Alternatively, use rule_23: A8 -> 8 10 A10 9. Could insert "7" before 8, and maybe delete the rest? No deletion. That yields "7 8 10 A10 9". This adds extra tokens 10 A10 9 after 8. Not good.

Thus seems tricky to get both patterns simultaneously given limited insertions. Perhaps there is a different parse for A22 that avoids needing "7 8". Let's revisit A22 parse. Previously we used A22 -> A7 17, and we set A7 -> 15 4 A9 (modified) to generate "15 4 11 A8 6 3". That produced "15 4 11 7 8 6 3". Wait we didn't need "7"? Wait we inserted "7" using rule_22 for A8. So that matches we needed to produce "7 8". That's inside A9->11 A8 6. So we needed A8 to produce "7 8". That required modification of A8's rule_22 to produce "7 8". That then conflicts with earlier usage.

But maybe we could change parsing for the first A8 (in A23) to not need "8 6". Let's see if we could have A7->7 A8 but produce "8 6" via insertion to A7 rule instead. For example, we could keep A8 as just "8", and insert "6" after A8 within the A7 production: modify rule_18: A7->7 A8 to be "7 A8 6". That would give "7 8 6". Indeed, if we modify rule_18 (A7->7 A8) to "7 A8 6" (inserting terminal 6 after A8), then A8 can be just "8". So we wouldn't need to modify A8 for that purpose. Let's consider.

Thus we could modify rule_18: A7 -> 7 A8 6 (inserting "6" after A8). That's a single insertion. Then A7->7 A8 6 yields "7 8 6". [A8 expands to "8" yields "7 8 6". Good.]

Then for the latter part, we want A8 to produce "7 8". That could be achieved by modifying rule_22 to "7 8". That's one insertion.

Thus we have consistent: in A23, we use A7 rule_18 with inserted "6" after A8, while A8 gives "8". In A22, we use rule_22 (modified) to give "7 8". Wait A22 uses A9->11 A8 6 3 after we inserted "3". In A9, we get "11" then A8's expansion "7 8", then "6" then "3". That yields "11 7 8 6 3" as needed. So that works.

Thus modifications:

- rule_18 (A7 -> 7 A8) modify to insert "6" after A8: "7 A8 6". (1 insertion)

- rule_22 (A8 -> 8) modify to "7 8". (1 insertion)

- rule_25 (A9 -> 11 A8 6) modify to "11 A8 6 3". (1 insertion)

That yields A22's needed sequence.

Now we check if we used rule_22 elsewhere for A8 in other contexts; we might need to check other uses.

We have other uses of A8: There's also A7->7 A8 for other A7 expansions we may use rule_18. In A7->7 A8 (rule_18) now with inserted "6". That covers all A7->7 A8 uses. That includes the earlier A7 in A23 (good). Note that there is also A7 in many other places we might use different productions (like rule_17, rule_16, and rule_19). So globally customizing rule_18 is safe.

Now, any usage of A8 for other productions still uses rule_22? Let's see: Where do we use A8 aside from A9? There is also rule_21: A8 -> 2 4 A9 3, and rule_23: A8 -> 8 10 A10 9. Are those used anywhere? Possibly not in our parse. Let's see: A8 appears at three places: A7->7 A8 (we used rule_18), A9->11 A8 6 (A8 as part of A9), and perhaps within other patterns? Let's search productions that mention A8:

- rule_18: A7 -> 7 A8
- rule_21: A8 -> 2 4 A9 3
- rule_22: A8 -> 8
- rule_23: A8 -> 8 10 A10 9
- rule_25: A9 -> 11 A8 6
- rule_27: A10 -> 5 A8

Also A10's production rule_27: A10 -> 5 A8

- rule_24: A8 -> 1
- rule_20: A8 -> 1

Wait we saw rule_20: A8 -> 1. Actually rule_20: A8 -> 1; rule_24: A8 -> 1. That could be used in other contexts. But we didn't use those.

Thus we need to ensure modifications don't break other needed expansions.

We will modify rule_22: A8 -> 8 -> add "7" before "8" (i.e., "7 8"). That changes all uses of rule_22, but do we need any other place where we rely on A8->8 producing just "8"? Not currently. In our parse after modifications, we have:

- A7->7 A8 (modified with inserted "6") uses A8 rule_22 (now "7 8") yields "7 7 8 6". Wait careful: A7->7 A8 6: A8 yields "7 8". So full expansion: "7 7 8 6". That's not desired. Indeed we need "7 8 6". If we modify rule_22 to "7 8", then A7->7 A8 6 yields "7 7 8 6". That's wrong (duplicate 7). So we need to adjust our plan.

We need A7->7 A8 to produce "7 8 6". If A8->8, we get "7 8 6". So we need A8->8 (no extra 7). That suggests we cannot modify rule_22 to "7 8". Instead we need another way to get "7 8" in A9's A8.

Thus we need to find a separate route to get "7 8". Perhaps we could use rule_20: A8->1 and insert "7 8" before 1 and after something? But then we get extra 1. However maybe we can have the extra 1 suppressed by using a rule that deletes it? Not possible.

Alternatively, maybe we can modify rule_23 (A8->8 10 A10 9) to produce just "7 8"? For that we could insert "7" before 8, and delete "10 A10 9"? Not possible. Could we insert only "7" at the beginning and then maybe insert epsilon after, or manipulate superfluous parts? Not allowed to delete terms.

But we can cause "10 A10 9" to yield unproductive terminals possibly matching needed tokens like "10" and "9"? Wait the target we need after "7 8" is a "6" (the terminal from A9). The "10 A10 9" would produce extra tokens "10 ... 9". That's not needed.

Thus we need to find a way to have A8 produce "7 8" using an existing rule but with insertions that preserve the tokens. Let's examine rule_20: A8->1. Insert "7 8" before and after? Insert before yields "7 8 1". After yields "1 7 8". Neither is "7 8". But the extra "1" might be fine if we can have that "1" be consumed by something else later? But the context expects only "7 8" then "6 3". There's no "1" between 8 and 6. So not good.

Rule_21: A8->2 4 A9 3. Insert something to get "7 8"? Insert "7" before "2"? and maybe modify A9 to produce something? Hard.

Possibly we could modify rule_28 (A11 -> 1) to include "7 8"? Not relevant.

Maybe we need an alternative parse of A22 that doesn't involve using A9->11 A8 6. Could we use a different production for A7 inside A22? In A22 we need to produce tokens "15 4 11 7 8 6 3". Instead of using A7->4 A9 (modified) and A9->11 A8 6 3, perhaps we can generate that using another structure: maybe use other productions from A7 that produce "7" as first token after "11"? Let's examine A7's other productions: rule_16: A7->1; rule_17: A7->4 A9; rule_18: A7->7 A8; rule_19: A7->10 A10 9.

We used rule_17 for the "15 4 ...". Indeed we inserted 15 before 4 into rule_17, giving A7->15 4 A9. That gave "15 4" prefix. Then A9 gave "11 A8 6 3". We need then "7 8" from A8. So need A8 to produce "7 8". That seems the only place to produce that "7 8". If we cannot get A8 to produce "7 8" within allowed modifications, perhaps we should find an alternative approach: for A9->11 A8 6 3, we could insert "7 8" before "11"? Actually we could modify rule_25 to include "7 8" before 11, making A9 produce "7 8 11 A8 6 3". But then we would need to adjust the preceding tokens accordingly.

Let's try to see if we can pick a different rule for A9 that yields needed pattern without needing A8 to produce "7". For A9 -> 1 (rule 24) cannot.

Or A9 -> 11 A8 6: we could insert "7 8" between 11 and A8 (or before 11). Option: modify rule_25 to be "11 7 8 A8 6 3". That would produce "11 7 8 <A8 expansion> 6 3". If A8 expands to nothing (i.e., epsilon) we could get "11 7 8 6 3". But A8 cannot be epsilon; it's a nonterminal that must produce something. However we could modify A8 to produce empty? Not possible; cannot delete its form.

Alternatively, modify rule_25 to "11 A8 7 8 6 3"? Not viable.

Is there another production for A9 that could be used? Only two productions: A9->1 (rule_24) and A9->11 A8 6 (rule_25). So we must use rule_25.

Thus we must get A8 to produce "7 8". Let's explore all A8 productions:

- rule_20: A8 -> 1
- rule_24: A8 -> 1 (duplicate)
- rule_21: A8 -> 2 4 A9 3
- rule_22: A8 -> 8
- rule_23: A8 -> 8 10 A10 9

We can modify any of these by insertion. A9 is a nonterminal that yields something (like "1" or "11 A8 6").

Goal: get A8 to expand to "7 8". Let's see if we can use rule_21: "2 4 A9 3". Insert "7" before "2"? yields "7 2 4 A9 3". If we can also make A9 produce "8" and maybe other stuff? Let's examine A9->... If we use A9->1 maybe we get "7 2 4 1 3"? That's not "7 8". If we use A9->11 A8 6, that yields "7 2 4 11 A8 6 3". That gives more tokens.

We could also insert "8" after "2"? Could produce "2 8 4 A9 3"? But we need "7 8" exactly.

Perhaps we could use rule_23: "8 10 A10 9". Insert "7" before the "8"? yields "7 8 10 A10 9". That produces "7 8 10 ... 9". Not just "7 8". But maybe we can make rest produce epsilon essentially by setting A10-> something that gives no terminals? Let's see A10 rules:

- rule_26: A10 -> 1
- rule_27: A10 -> 5 A8

Both produce terminals (1 or 5 plus expansion). So can't get empty.

Thus rule_23 can't give just "7 8". Too many tokens.

Now rule_20/24: "1". Insert "7 8" before "1"? "7 8 1". That's three tokens. Not good.

But maybe we can modify rule_20 to insert "7 8" after "1": "1 7 8". That yields "1 7 8". Not "7 8". Could we then have preceding token "11" from A9 and then "1 7 8 6"? That would yield "11 1 7 8 6". Not correct.

Thus the only way seems to get "7 8" from A8 is to modify rule_22 to "7 8". But that conflicts with earlier use for A23 because A7->7 A8 needed to produce "7 8 6". However we could modify A7 rule_18 accordingly: initially A7->7 A8; if A8 is now "7 8", we could modify A7->7 A8 6 to produce "7 7 8 6". To avoid duplication, perhaps we can modify A7->7 A8 to produce "13"? Not right. Let's consider alternative: Use different A7 production for the early A7 (in A23) that does not rely on A8. For A23's A7, we can use rule_16: "A7 -> 1" then insert extra terminals to produce "7 8 6"? Wait A7->1, we could insert "7 8 6" before or after "1". Insert "7 8 6" before "1" yields "7 8 6 1". That would give "7 8 6 1" as part of A23. However we need "7 8 6 1"? Actually the first part of target is "7 8 6 1". That's exactly "7 8 6 1". So maybe we could instead have A7 produce "7 8 6 1" via inserting those terminals before the final "1". Then we wouldn't need A7->7 A8 and A8 modifications. Let's examine this.

In A23, we have A7 (first) A21 18 A24. We currently had A21->1 (giving token 1), and A24->A7 -> 1 (giving token 1). But if we make the first A7 produce "7 8 6 1", then A21 would still need to produce something? Actually our initial token sequence for A23 needed "7 8 6 1 18 1". We have A21->? The token "1" after "6" is from A21, which we set to 1 using rule_56. That seems okay. But if we make A7 produce "7 8 6 1", then we would have extra "1" before A21 token 1, leading to "7 8 6 1 1 18 1". That's not correct. However maybe we could change A21 to produce something else (like not 1). Let's examine: A21 options are 1 or 16. We need after A7's expansion we have "1" from A21 (pos4) before 18. And after 18 we need "1". So we need exactly a 1 token coming from A21, not more. So we shouldn't let A7 produce a 1; otherwise extra 1 appears.

Thus using A7->1 with insertions might need to ensure that the inserted tokens include "7 8 6" before the 1, making A7's expansion "7 8 6 1". Then A21 could be something else (maybe using rule_57 to produce 16). That would give "7 8 6 1 16 18 1"? Not correct. So not viable.

We could keep A7->7 A8 (or other), but need to get "7 8 6". Perhaps easier approach: modify rule_18 to be "7 8 6" directly, i.e., insert "8" and "6" but drop A8? Actually we cannot delete A8. Must keep A8. But we could insert "8" before A8 and "6" after A8, and then modify A8 to be epsilon? Not possible. But we could make A8 produce empty? Not allowed.

Thus best is to accept modifying rule_22 to "8". Keep original to produce "8". Then modify rule_18 to "7 A8 6" (insertion of 6 after A8). This gives A7->7 A8 6. A8->8 (unchanged). That yields "7 8 6". Perfect. No insertion into A8 needed for first use. This matches target.

Now for A9's A8, we need to produce "7 8". However rule_22 is still "8". So we need different method for A8 in that context. Perhaps we can use rule_20 or rule_24 with insertion.

Let's examine using rule_20: A8 -> 1. We could insert "7" before the 1, and maybe also insert "8" after the 1, making A8 -> 7 1 8. That yields "7 1 8". We're looking for "7 8". Could we make the extra "1" match something in the overall sequence? Let's see context: we are inside A9->11 A8 6 3. After "11", we want "7 8" then "6 3". If A8->7 1 8, we get "11 7 1 8 6 3". That's "11 7 1 8 6 3". The target is "11 7 8 6 3". There's an extraneous "1". Perhaps we could assign that "1" to be consumed by the preceding A7->15 4 A9? Wait that is part of same A7, which currently is "15 4 A9". The A9 expansion includes that "1". Could we incorporate that "1" as part of the sequence "15 4 ..."? Let's analyze.

Our target for that region is "15 4 11 7 8 6 3". This is indeed A7->15 4 A9 (which yields "15 4" then A9). So A9 must produce "11 7 8 6 3"? Wait we want after "15 4" we have "11 7 8 6 3". So A9 may produce "11 7 8 6 3". Using rule_25 with modifications.

If we modify rule_25 to be "11 A8 6 3" (original plus insertion of 3 is already needed). If A8 produces "7 8", we get "11 7 8 6 3". Good.

If we use A8->7 1 8 (via rule_20 with insertions), A9: "11 7 1 8 6 3". That's extra "1". So not match.

Thus we need A8 produce exactly "7 8". As before, we cannot use rule_22, as that yields "8". To get "7 8", perhaps we can use rule_20 (1) with insert before "1" of "7 8"? That would be "7 8 1". That's "7 8 1". A9 would be "11 7 8 1 6 3"? Actually A9->11 A8 6 3: after A8 we have 6 3. So if A8 = "7 8 1", we get "11 7 8 1 6 3". That's "11 7 8 1 6 3". There's extra "1" before 6. But target after "11 7 8" is "6" (immediate) not "1". So not good.

Maybe use rule_21: A8 -> 2 4 A9 3, with insertions to produce "7 8"? Could we produce "7 8" by using A8->2 4 A9 3 and then making A9 produce epsilon and adjusting? Let's see: "2 4 A9 3". Insert "7" before "2"? yields "7 2 4 A9 3". Then we need to produce "8" after? Insert "8" after "4"? Could do "2 4 8 A9 3"? That's "2 4 8 A9 3". Still not just "7 8". The A9 part would produce something maybe "empty"? Not possible.

Thus direct production generating "7 8" seems only possible using rule_22 with insertion of "7" before "8". That yields "7 8". Let's consider: we can modify rule_22 to "7 8". That works for A9's A8. But then for A7's A8 (the first instance) it would also become "7 8". However we can adjust rule_18 differently: use A7->7 A8 6 but now A8 yields "7 8". So A7 yields "7 7 8 6". That adds an extra 7 we don't want. But perhaps we can adjust rule_18 to "A7->7 6"? No; can't delete A8.

Alternatively, we could instead use a different A7 production for the first usage that doesn't involve A8; use e.g., A7->4 A9 (but produce "7 8 6"? Not directly). Let's explore.

Goal for A23's first A7: need to produce "7 8 6". A7 could use rule_16: A7->1, with insertions "7 8 6" before "1"? That yields "7 8 6 1". But then results in "7 8 6 1" which is okay only if we set A21 to something else? Actually after initial tokens we need "7 8 6 1". But note currently after A23 we have A21 then 18 etc. If we incorporate the "1" into A7, maybe we reduce need for A21 to produce "1". Let's examine if we can reassign tokens: we need "7 8 6 1 18 1". If A7 produces "7 8 6 1", then we need after A7 (i.e., after A23's A7) to have A21 produce something else (maybe empty? but must produce a token). A21 can produce either "1" or "16". If we set A21 to produce maybe something that can be omitted? We need only one "1" before "18". But we already have a 1 from A7. Actually target "7 8 6 1 18 1". There are two "1"s: one at position4, one at position6. We could map the first "1" (pos4) from A7's inserted "1" and then A21 could be something else (maybe "18"? No). But A21 must produce a token. So if A7 gives "7 8 6 1", we already have the 1. Then A21 could be something else like produce "16"? That would be extra token not required. So not suitable.

If A7 yields "7 8 6" (no 1), then A21 yields "1" to satisfy the first 1. That's our earlier approach. So we need A8->8 for A7's usage, not "7 8". So rule_22 must remain "8". Thus we cannot modify rule_22 to "7 8". So we need an alternative for A9's A8.

Thus we need to produce "7 8" from A8 using some other rule. Let's examine all possibilities again.

Potential path: Use rule_20 (A8->1). Insert "7" before "1" and "8" after "1": yields "7 1 8". That's not correct.

Use rule_24 (duplicate of rule_20). Same issue.

Use rule_21 (2 4 A9 3). Could insert "7" before "2" and "8" after "4"? That yields "7 2 4 8 A9 3". Then we need A9 to equal epsilon (cannot) or something that yields no terminals. Not possible.

Use rule_23 (8 10 A10 9). Could insert "7" after "8"? yields "8 7 10 A10 9"? That's "8 7 ..." Not correct order.

But we could modify rule_23 to insert "7" before "8": "7 8 10 A10 9". Then perhaps we can also insert something to get rid of "10 A10 9"? If we insert something that makes them matched with tokens later? But after A9 we need "6 3". There's no "10 A10 9" tokens. Hmm.

Alternatively, maybe we can avoid using A9's A8 at all by adjusting A9->11 A8 6 to produce "11 7 8 6 3" with A8 being something else that yields "7"? Wait if A8 produces just "7", then A9 yields "11 7 6 3". That's missing "8". Not correct.

Alternatively, we could note that we can insert terminals not only in the target A8 rule but also we could insert terminals before A9's A8 in rule_25. For instance, modify rule_25 to "11 7 8 A8 6 3". Then A8 could be something minimal (like "1") and we get "11 7 8 1 6 3". Still extra 1. But maybe we can set A8 to "epsilon"? Not possible.

We could use rule_25 to insert both "7" "8" before A8, as above, but then we also need A8 to produce something that might be "zero"? Not possible.

Alternatively, we could make A9 generate "11 7 8 6 3" directly by inserting both "7 8" and also a "6"? Wait we already have "6" from rule. So we only need "7 8". Inserting those before A8 yields "11 7 8 A8 6 3". If A8 expands to empty (not possible) or some way to produce empty via something like "A8 -> 1" then we have "11 7 8 1 6 3". But extra "1". Could we instead modify rule_25 to "11 7 8 6 3"? i.e., replace A8 with nothing, but we cannot delete A8. We could modify rule_25 to "11 7 8 6 3" by inserting "7 8" before the existing "6", and also deleting A8? Not allowed.

Thus maybe we need to modify A9 differently by using alternative production (A9->1) but then modify A9->1 to include all needed tokens "11 7 8"? But rule 24: A9->1; we could insert "11 7 8" before that 1: becomes "11 7 8 1". But then we also need trailing "6 3"? Wait A9->1 is used inside A7->15 4 A9. So A9 expansion would be "11 7 8 1". That would generate "11 7 8 1". Then after A9 we have the suffix from rule 17 (our inserted "3" after 6?), but we used rule_25 for that. Let's see: A7->15 4 A9, A9 maybe we could redirection. However A9->1 is a distinct production that yields only 1 terminal. We could insert required terminals around 1. For example, we could modify rule_24: A9->1 to "11 7 8 1". Insert "11 7 8" before the "1". Then A9 would yield exactly "11 7 8 1". Then after that, we have A7's suffix? Wait A7->15 4 A9 (we inserted "15" before "4"). Nothing else after A9; the A7 expansion ends with A9. Then from A22, after A7 we have A22's terminal "17". Actually A22 is A7 17, not A7 plus something else. Wait sequence: A22 -> A7 17. So the A7 here is after A21 18: It's a separate A7, not the same as A7 in A23 etc. In A22, we need to generate "15 4 11 7 8 6 3". Wait double-check: A22 appears after A21 and before A19. A22->A7 17. So A22 contributes A7's expansion and then 17. Actually earlier we derived: A22 needed to generate "15 4 11 7 8 6 3". But note A22's terminal 17 appears after A7. Since target has "...3 17...". Indeed after token "3" we have "17". So the part before 17 must be from A7. So yes, A7's expansion must be "15 4 11 7 8 6 3". So we must get that from A7.

Thus we need A7 (some production) to produce that long string. We've been using A7->15 4 A9, with A9->11 A8 6 3.

Thus A7's expansion contains A9's expansion. This seems central.

Thus we need to solve the issue of finding A8 to generate "7 8". Options: we could modify A9's rule to give "11 7 8 6 3" by adjusting rule_25 rather than needing A8 produce "7 8". Let's attempt to modify rule_25: "A9 -> 11 A8 6" to produce "11 7 8 6 3". But A8 would still produce something after inserted "7 8"? Actually we can insert "7 8" before the A8, and also we can maybe insert nothing after? But still A8 adds its own token(s). Could we modify A8 to produce an epsilon? Not possible.

But perhaps we can modify rule_25 to "A9 -> 11 7 8 6 3" and also insert a nonterminal (like a new A8) but we can't delete. Could we insert "7 8" after 11, then also insert something to swallow the A8? No, because A8 cannot vanish; it's a nonterminal that will produce at least one terminal (since its productions produce at least one terminal). So we need to manage that.

Thus it seems the only viable way is to have A8 produce just "7 8". This requires rule_22 = "7 8". So we must adopt that.

Now we need to adapt the first occurrence of A8 (in A23's A7->7 A8) to still produce "7 8 6". If A8 becomes "7 8", then A7->7 A8 6 yields "7 7 8 6". That's extra "7". But we could modify A7's rule_18 to "A7 -> 7 A8" (i.e., no extra 6) and then incorporate the "6" elsewhere (perhaps by inserting "6" after A7 in A23's RHS). Let's explore.

We have A23's first A7 (call it A7_1). Its production is currently rule_18 (used): A7 -> 7 A8 (original). We previously inserted "6" after A8 (making it "7 A8 6"). But now if we want to maintain A8->7 8, then A7->7 A8 yields "7 7 8". That's "7 7 8". But we need "7 8 6". Maybe we can modify rule_18 to add insertions before A8 to remove extra 7, etc. Let's attempt to rework.

Consider using a different production of A7 for the first A7 (maybe rule_16: A7->1 with insertions). Example: use rule_16: A7->1, then insert "7 8 6" before the "1"? That gives "7 8 6 1". Then after A7 we have A21, which we currently set to produce "1" at position4. That would produce "7 8 6 1 1". That's an extra "1". But we could set A21 to produce "something else"? A21 can produce "16". That would give "7 8 6 1 16". But target is "7 8 6 1 18". Not match.

Alternatively, we could change A21's production to produce "18"? Actually A21 only produces 1 or 16. So can't.

Thus better to create A7 producing "7 8 6" directly (no extra "1"). Could we have A7 produce "7 8 6" using rule_18 but with A8 producing "8" and then insert "6"? Already we considered that. But we also have to adjust later uses.

Thus perhaps we need to keep A8 as "8" for that context, and something else for the later context. However we cannot have two simultaneously different definitions for A8. Could we differentiate by using a different rule for the later A8? Yes, we can choose a different production for the later A8 that can be modified to produce "7 8". For example, use rule_20 (A8->1) and modify it to produce "7 8". How to get "7 8"? Starting from "1", we could insert "7" before and "8" after, but would also have the "1". That would give "7 1 8". Not right. But we could modify rule_20 to be "7 8 1"? Insert before and after yields "7 8 1". That would produce "7 8 1". If we then modify rule_25 to include "6 3" after A8, we would have "11 7 8 1 6 3". There's an extra 1. But perhaps we could also modify rule_25 to delete that 1? Not allowed. Could we also modify rule_20 to "7 8"? We cannot delete "1". So can't.

Alternatively, perhaps we could use rule_24 (duplicate) but same problem.

Thus the only way to get A8 produce "7 8" without extra symbols is to modify rule_22 to "7 8". Fine.

Now the conflict with A23's A8 is that we need it to be "8". But we could use a different rule for A8 there: not rule_22, but maybe rule_21: "2 4 A9 3"? We could heavily insert to get "8". For example, start with "2 4 A9 3". Insert before "2" a "8"? That yields "8 2 4 A9 3". Need A9 to produce epsilon or something we can adjust. Not good.

Alternatively, rule_23: "8 10 A10 9". We could insert terminals to make this just "8"? Could we insert "epsilon"? No. But we could insert terminals that may later cancel? Not possible.

Thus better to keep A8 as "8" for A23's use. So rule_22 is currently "8". We need a separate A8 with "7 8". This seems impossible given we only have four production rules for A8 and each cannot be context-specific.

But maybe we can use A9->11 A8 6 to produce "11 7 8 6"? If we set A8 via rule_22 to "7"? But rule_22 currently "8". Could modify rule_22 to "7"? Actually A8->7? Insert "7" before "8"? That gives "7 8". But also includes "8". Maybe we can modify A8->8 to "7"? Insert before or after 8 can't delete 8. So we cannot get "7" only.

Thus we may need to re-evaluate approach. Let's re-evaluate the parsing of the sub-sequence "15 4 11 7 8 6 3". Perhaps there is an alternative derivation that doesn't require A8 to produce "7 8". Possibly we can achieve "15 4 11 7 8 6 3" via different sequence of productions, perhaps using other nonterminals.

Let's see other productions that could produce "7 8". For example, A7->7 A8 (we know A8 could be "8") yields "7 8". Actually if A7->7 A8, and A8->8, then we get "7 8". Perfect! But earlier we needed A8 "8". That's okay. So maybe we can use A7 (somewhere) to produce "7 8". In our parsing we used A9->11 A8 6 3. But perhaps we can modify that to use A7 instead of A8, constructing "11 7 8 6 3". Let's examine: A9->11 A8 6; we can replace A8 with a different nonterminal that produces "7"? Actually A8 is fixed. But maybe we can replace A9 with alternative production A9->1? Not helpful.

But maybe we could adjust A9's rule to produce "11 A7 6 3". That would require inserting "A7" nonterminal into RHS, but we cannot insert nonterminals—only terminals. So no.

Thus A9 must have a nonterminal A8 inside; we can't replace with A7.

Thus the only approach to get "7 8" inside A9 is for A8 to produce "7 8". So we must adopt rule_22 to produce "7 8". Then we need to adapt the A23's first usage accordingly.

Now, maybe we can avoid using A8 altogether for the first part of A23 by using a different A7 production that doesn't involve A8. For example, we could use A7->4 A9 (rule_17) after inserting appropriate terminals to make "7 8 6". Let's see: A7->4 A9 with insertions: we could insert "7 8 6" before "4"? The RHS would become "7 8 6 4 A9". Then we need to make A9 produce minimal (maybe empty) to just have "7 8 6". But A9 will produce at least one terminal, unless we make A9 produce something that can be inserted as nothing? No.

What about A7->10 A10 9 (rule_19)? Could insert "7 8 6" before "10"? ... Not promising.

Thus using A7->1 with insertions to produce "7 8 6 1"? Already considered.

Maybe we can change A23's A7 to use rule_18 with modification to produce "7 8" directly, and then incorporate "6" from elsewhere, maybe from A21? A21 currently produces 1, not 6. Could modify A21 to produce "6"? A21 productions: 1 or 16, can't produce 6 unless we add insertion to A21's rule_56: "A21 -> 1", we could insert "6" after 1 to get "1 6". But token order would be "1 6". We need "6" before 1 (the "6" is at position3, before the "1" at position4). Actually after "7 8" we need "6" then "1". So we could have A7 produce "7 8", and A21 produce "6 1". To achieve that, modify rule_57: A21 -> 16? Not relevant. Use rule_56: "A21 -> 1". Insert "6" before the 1 so rule becomes "6 1". That yields "6 1". Good. So we can have A7->7 A8 (rule_18 unchanged) where A8 via rule_22 yields "8"? Wait rule_22 currently "8". So A7->7 A8 yields "7 8". Great. Then A21->1 with insertion "6" before => "6 1". Thus A23 yields "7 8 6 1 18 ...". That matches the first five tokens: 7,8,6,1,18. Then A24->A7 must give "1" (the final token before next). A24->A7 uses A7 matching either rule_16 (1) or etc. A7 rule_16 yields "1". So that yields token7 = 1. So overall we get "7 8 6 1 18 1". Perfect.

Thus we can achieve the start using the following modifications:

- Keep rule_22 as "8" (no insertion for this). No modifications needed for rule_22.

- Modify rule_18 (A7->7 A8) to possibly add nothing (no insertion needed), because A8 yields "8". So A7 yields "7 8". Works.

- Modify rule_56 (A21 -> 1) to insert "6" before the "1", i.e., "A21 -> 6 1". That's one insertion.

Thus where to get "6" token after "8"? Yes, the inserted "6" will be part of A21, not A3. The sequence becomes: A7->7 A8 = "7 8". Then A21->6 1 yields "6 1". So overall pattern = "7 8 6 1". Then A23's terminal "18". Good.

Thus we can avoid insertion in A8. Good.

Thus we can revert modifications we earlier invented for A8; just modify A21.

Now check later A22 parsing. As earlier we need "15 4 11 7 8 6 3". This uses A7->? Actually A22->A7 17: A7 must produce "15 4 11 7 8 6 3". Let's see if we can accomplish that with changes consistent with rules.

We earlier used A7->15 4 A9 (modified rule_17) and A9->11 A8 6 3 (modified rule_25) and A8->7 8 (modify rule_22). This demands rule_22 to produce "7 8". But note that rule_22 is also used in A23's A7->7 A8: that would produce "7 7 8". Wait A7 uses rule_18, not rule_22. So A23 uses rule_18, which has A8 too. So A8 appears there as well (in A7->7 A8). That uses rule_22 again. If we modify rule_22 to "7 8", then A7->7 A8 would produce "7 7 8". That's not what we want (previously we rely on A7->7 A8 being "7 8" only). But we can modify A7->7 A8 rule (rule_18) to insert something that deals with the extra "7"? For example, modify rule_18 to be "7 A8" but we might need to adjust insert to remove the extra 7. Actually if A8 -> "7 8", then A7->7 A8 yields "7 7 8". If we insert a deletion, not allowed. But we could modify rule_18 to "7 8 A8"? No.

Better to adjust rule_18 to "A8" only? But can't delete 7. Or we could have rule_18 use insertion "6" after A8? That yields "7 7 8 6". Not good.

Thus we need to avoid using rule_18 for the A7 in A23 or find a different way to generate "7 8". Perhaps we can generate "7 8" using a different rule for A7 in that context, like rule_16 (A7->1) with insertions to produce "7 8". Let's attempt.

Goal for A23's first A7: produce "7 8". Choose rule_16: A7->1. Insert "7 8" before the "1" (insert before "1", then maybe we keep the 1? That yields "7 8 1". That gives extra "1". However we have A21->6 1, which gives "6 1". So we would have "7 8 1 6 1..." which mismatched (target is "7 8 6 1").

But maybe we could insert "7" before the "1" and "8" after the "1". That yields "7 1 8". Then we could have A21 produce "6". But A21 would only produce a terminal if we produce "6"? Actually A21->16 or 1. Not helpful.

Thus maybe we can try to use rule_19: A7->10 A10 9. Could we insert "7 8" before or within to produce exactly "7 8"? Not likely.

Thus the simplest is to keep rule_22 as "8", and get "7 8" from A7->7 A8 (with A8->8). That's exactly "7 8". So that part works.

Thus we need to ensure that rule_22 remains "8". But then later we need A8 to produce "7 8". That's contradictory. But maybe we can avoid using A8 in A9's expansion. Let's revisit A22's parse: There might be another way to produce "11 7 8 6 3" without A8 producing "7 8". Suppose we could produce "11 7" using one A9 expansion, and "8 6 3" using subsequent productions. But A9 only appears inside A7->... etc.

Let's re-evaluate A22's target substring: "15 4 11 7 8 6 3". Is there any other path? Let's search other nonterminals that can produce those tokens.

We have many nonterminals A15, A16, etc which produce similar patterns.

Look at rule_40: A15 -> 4 A16 3. That's similar "4 ... 3". But we have "15 4" at start, not "4".

Check rule_47: A17 -> 2 4 A16 3. That's "2 4 ... 3". Not matching.

Check A11: A11 -> 4 A12 3. Also "4 ... 3". A12 also can be "1" or "11 A13 6". So A11 -> 4 A12 3 could give "4 1 3" or "4 11 A13 6 3". A13 can produce "1", "2 4 A12", "8 A14". So could we produce "4 11 7 8 6 3"? Let's try: Use A11 -> 4 A12 3, with A12 -> 11 A13 6 (rule_33). Then we have "4 11 A13 6 3". If A13 -> 8 A14 (rule_36), that yields "8 A14". So we get "4 11 8 A14 6 3". A14 can produce "1" or "5 A13 6". This seems not yield "7". So not the pattern.

Alternatively, A11 -> 7 A13 6 (rule_30). That yields "7 A13 6". If A13 -> 8 A14 (rule_36) yields "8 A14". So A11->7 8 A14 6. If A14 -> 1, we have "7 8 1 6". Not "7 8 6". Could add insertion "6"? no.

Alternatively, A11 -> 7 A13 6, with A13->2 4 A12, maybe produce "2 4 something". Not.

Thus that path seems not directly "11 7 8 6 3".

Let's examine A15 productions: A15 -> 4 A16 3 (rule_40). In order to get "15 4 ...", we would need A15 produce "4 ... 3". But we need "15 4 ..." which could be achieved by inserting "15" before rule_40's RHS.

Thus for A22's A7 we could use A7->15 A15 (some rule?), i.e., A7->? Actually A7's productions don't include A15. However we could generate "15 4 ..." via A7->4 A9 (rule_17) with insertion "15" before "4". That's what we did. Then A9 produces "11 A8 6" with insertion "3" after 6.

Thus we still need A8 to be "7 8". That's the problematic part.

But perhaps we could make A9 produce "11 7 8 6 3" via using a different production for A9 (like A9->1) and then using insertions to insert "11 7 8 6 3" around the "1". That would be: modify rule_24: A9->1 to "11 7 8 6 3 1" (but ordering). Actually we could do: "11 7 8 6 3 1". However A9 is then followed by the suffix after A9, which is nothing in A7 RHS (since A7->15 4 A9, then then A9->...). So A7 final after insertion would produce "15 4 11 7 8 6 3 1". That's an extra "1" we don't want. Could we instead insert after 1, not before? Like "1 11 7 8 6 3"? That yields "1 11 7 8 6 3". Not correct.

But we could also modify rule_24 to "1", but insert before "1": "11 7 8 6 3". That's "11 7 8 6 3" before "1". Then after A9 we get "1"? Actually RHS would be "11 7 8 6 3 1". That includes the 1; we can't drop that. Not okay.

Alternatively, use rule_25 with modifications: Insert "7 8" earlier (as we attempted) but keep A8->8. However we can insert "7" before A8 and "8" after, but still have the "8" from rule_22? Let's try.

Goal: we want A9 to produce "11 7 8 6 3". We have rule_25: "11 A8 6". If we modify this rule to "11 7 A8 6 3". Then A8 must produce "8". Then sequence is "11 7 8 6 3". Indeed if A8->8, we get "11 7 8 6 3". This works! Let's examine:

- Insert "7" after "11" (before A8).
- Insert "3" after "6".

Thus modified rule_25: "A9 -> 11 7 A8 6 3". Yes.

Now, does A8 need to produce "8"? Yes, we keep rule_22 as "8". So A8 yields "8". Then A9 yields "11 7 8 6 3". Perfect! This matches the needed substring.

Thus we needed 2 insertions in rule_25: "7" after "11" and "3" after "6". Wait we already needed "3" after "6". So that's one insertion. Actually earlier we considered we needed to insert "3" after "6" (that's 1). Now we also need to insert "7" after "11". That's another insertion. So total 2 insertions for rule_25.

Thus, with this change we don't need to modify A8. Good, we can keep rule_22 as original "8". So we have no conflict.

Thus A22's parse: Use A7->15 4 A9 (modified rule_17 to insert "15") => "15 4 <A9 expansion>"

A9 via modified rule_25: "11 7 A8 6 3". A8 -> "8". So overall A7 yields "15 4 11 7 8 6 3". Good.

Thus A22 = A7 17 yields "15 4 11 7 8 6 3 17". Exactly matches tokens pos15-22 (pos15=15, pos16=4, pos17=11, pos18=7, pos19=8, pos20=6, pos21=3, pos22=17). Perfect!

Thus we have solved A22 with insertions: modification of rule_17 (A7->4 A9) and rule_25 (A9->11 A8 6). Good.

Now we need to adjust other parts accordingly.

Now we continue parse after A22. After A22 expansion we have consumed tokens up to position22. Next is A19 at position23 onward.

We have k tokens left: position23=1, 24=13, 25=1, 26=12, 27=1, 28=14.

As earlier, we decided to use A19 -> A7 13 with A7->1 (rule_16) no insert, and A21? Actually A19 produce "1 13". That matches positions 23 = 1, 24 = 13.

Then A2 must produce "1 12". A2->A7 (rule_2) with A7 possibly using rule_16 (1) plus insertion "12". Let's modify rule_2 to insert "12" after A7.

Thus A2 yields "1 12". That matches positions 25 = 1, 26 = 12.

Finally, A20 -> A7 14, with A7->1 (rule_16) yields "1 14". That matches positions 27 = 1, 28 = 14. Perfect.

Thus we need modifications:

- Insert "6" before "1" in rule_56 (A21->1) to produce "6 1". (1 insertion)
- Insert "15" before "4" in rule_17 (A7->4 A9) to produce "15 4 A9". (1 insertion)
- Insert "7" after "11" and "3" after "6" in rule_25 (A9->11 A8 6). Actually need to insert "7" after "11", and "3" after "6". That's 2 insertions.

But wait we need to check: rule_25 currently "A9 -> 11 A8 6". It already has 11, A8, 6. To produce "11 7 A8 6 3", we add "7" after 11 and "3" after 6. That's 2 insertions.

- Insert "12" after A7 in rule_2 (A2 -> A7 12). That's 1 insertion.

Thus total insertions: 1 (rule_56) + 1 (rule_17) + 2 (rule_25) + 1 (rule_2) = 5 insertions.

But earlier we considered also maybe need to modify rule_18? Actually we have not inserted anything there now because we rely on A7->7 A8 (unchanged), A8->8. But we also earlier needed A21 to produce "6 1". We'll insert in rule_56.

Check everything else: Are any other missing terminals needed? Let's verify entire derivation step-by-step with modifications.

Define new grammar modifications:

- rule_2: "A2 -> A7 12" (insert 12 after A7).
- rule_17: "A7 -> 15 4 A9" (insert 15 before 4).
- rule_25: "A9 -> 11 7 A8 6 3" (insert 7 after 11, insert 3 after 6).
- rule_56: "A21 -> 6 1" (insert 6 before 1).

We also might need to adjust rule_18? Let's double-check our parse:

A1 -> A23 A3 A21 A22 A19 A2 A20

Now step through:

1. A23 -> A7 A21 18 A24

- For A23's first A7, we need to produce "7 8" using rule_18: A7 -> 7 A8 (no insert). A8 uses rule_22: "A8 -> 8" (no insert). So A7 yields "7 8". Good.

- A23's A21 (the second non-terminal in RHS after A7) is? Wait rule_1 includes A23 A3 A21 A22 A19 A2 A20. A23 includes inside it A7 A21 18 A24. So we need to differentiate A23's internal A21 (call it A21_1) from the outer A21 (call it A21_2). A23's internal A21 must produce "6 1"? Indeed we need "6 1". Let's confirm: After A23's A7 (which gave "7 8"), we need token 3=6, token4=1 before the terminal 18. So internal A21 must generate "6 1". So modify rule_56 (the only rule that yields "1") for that internal A21. But note we also have outer A21 (the one after A3) that also uses the same non-terminal A21. It also will use rule_56 possibly, unless we assign a different production (rule_57: A21 -> 16) for that instance. Let's see: We need outer A21 to generate what token? After A3 we have A21 (outer) at position? Let's chart.

Sequence:

- A23 -> part: tokens 1-6: "7 8 6 1 18 1". Let's break it: A7 -> "7 8". internal A21 -> "6 1". then terminal 18. then A24 -> A7 (which yields "1").

Thus A23 as a whole yields: A7 (7 8), A21 (6 1), 18, A24 (A7->1). So after A23 we have consumed "7 8 6 1 18 1". Great.

Thus internal A21_1 uses rule_56 with insertion "6" before 1. Good.

Now A24 -> A7. Which A7 will we use? It can use rule_16: A7 -> 1 (just 1). That's fine. No insertion needed.

Thus after A23 we go to A3 (next nonterminal).

- A3: we chose rule_7 with insertion of 19 before 10: modify rule_7 (currently "A3 -> 10 A6") to "A3 -> 19 10 A6". No other modifications.

Now A6's expansion:

We need to produce tokens "5 7 2 6 9". A6 -> 5 7 A5 6 with insertion of "9" after 6. Let's modify rule_15: "A6 -> 5 7 A5 6 9". Also we need A5->2 (rule_11) to produce "2". So A6 yields "5 7 2 6 9". Good.

Thus after A3 we have "19 10 5 7 2 6 9" (positions 7-13). Next nonterminal after A3 is A21 (outer). This outer A21 must generate token14=16. We can use rule_57: A21 -> 16 (no insertion). No need for rule_56 here, as we used rule_56 only for internal A21_1. That's okay: A21 can be either 1 or 16. So outer A21 uses rule_57 (no modification). Good.

Now next is A22. As above, A22 -> A7 17. Use A7 as per rule_17 modified. A7->15 4 A9 yields "15 4 <A9 expansion>". A9 modified to "11 7 A8 6 3". A8 remains "8". So A7 yields "15 4 11 7 8 6 3". Then A22 adds terminal "17". So A22 yields tokens positions 15-22: "15 4 11 7 8 6 3 17". Good.

Now next is A19. Use rule_52: "A19 -> A7 13". Use A7 -> 1 (rule_16). So A19 yields "1 13". That's tokens pos23-24. Good.

Now A2: use rule_2 modified to "A2 -> A7 12". Use A7->1 (rule_16). So yields "1 12". That's tokens pos25-26. Good.

Now A20: rule_55: A20 -> A7 14. Use A7->1 (rule_16). Yields "1 14". That's tokens pos27-28. Good.

Thus all tokens are derived correctly.

Now verify we haven't used any other symbols incorrectly.

Now the modifications summary:

- rule_2: add terminal 12 after A7.
- rule_56: add terminal 6 before 1.
- rule_7: add terminal 19 before 10.
- rule_15: add terminal 9 after 6.
- rule_17: add terminal 15 before 4.
- rule_25: add terminal 7 after 11 and terminal 3 after 6.

That's 6 modifications. Wait we also need to modify rule_15 (A6->5 7 A5 6) to add "9". Yes 1 insertion. So total modifications = 1 (rule_2) + 1 (rule_56) + 1 (rule_7) + 1 (rule_15) + 1 (rule_17) + 2 (rule_25) = 7 insertions? Let's count:

- rule_2: 1
- rule_56: 1
- rule_7: 1
- rule_15: 1
- rule_17: 1
- rule_25: 2
Total = 7.

But we earlier counted 5 (excluded rule_15 and rule_7). Indeed we need to include the insertion of 19 and insertion of 9 as we earlier noted. So total is 7.

Now check rule_15 after insertion: Original rule_15: "A6 -> 5 7 A5 6". Insert "9" after "6". So RHS becomes "5 7 A5 6 9". OK.

Now check rule_7 after insertion: Original rule_7: "A3 -> 10 A6". Insert "19" before "10". So "19 10 A6". Good.

Check rule_56: original "A21 -> 1". Insert "6" before "1": "6 1". Good.

Check rule_2: original "A2 -> A7". Insert "12" after A7: "A7 12". Good.

Check rule_17: original "A7 -> 4 A9". Insert "15" before "4": "15 4 A9". Good.

Check rule_25: original "A9 -> 11 A8 6". Insert "7" after "11", and "3" after "6": "11 7 A8 6 3". Good.

Now verify there are no other needed modifications. Let's run through entire derivation with modifications to ensure matches exactly.

### Full Derivation step-by-step

Start: A1

Rule 1: A1 -> A23 A3 A21 A22 A19 A2 A20

Sequence to derive: concatenate expansions.

#### Derivation for A23:

Rule_52 (and others). Pick rule_52: A23 -> A7 A21 18 A24.

- A7 (first) uses rule_18: (unmodified) A7 -> 7 A8.

- A8 uses rule_22: A8 -> 8. (unmodified)

=> yields "7 8".

- A21 (internal) uses rule_56 modified: "6 1". => yields "6 1".

Now we have so far "7 8 6 1". Next, "18" literal: yields "18". So far "7 8 6 1 18".

- A24 uses rule_60: A24 -> A7.

A7 (for A24) uses rule_16: A7 -> 1. => yields "1".

Thus A23 yields: "7 8 6 1 18 1". Good.

#### Derivation for A3:

Choose rule_7 modified: "A3 -> 19 10 A6".

- So adds "19 10".

- A6 uses rule_15 modified: "A6 -> 5 7 A5 6 9".

Now expand A5: choose rule_11: A5 -> 2.

Thus A6 yields "5 7 2 6 9". Total A3 yields "19 10 5 7 2 6 9". Good.

#### Derivation for outer A21:

We need token "16". Choose rule_57: A21 -> 16. => yields "16". Good.

#### Derivation for A22:

Rule_58: A22 -> A7 17. So yields expansion of A7 then "17".

Now A7 uses rule_17 modified: "A7 -> 15 4 A9".

- Emits "15 4".

- A9 uses rule_25 modified: "A9 -> 11 7 A8 6 3".

Now A8 uses rule_22: "8". So A9 yields "11 7 8 6 3". So A7 yields "15 4 11 7 8 6 3". Then A22 adds "17". So total "15 4 11 7 8 6 3 17". Good.

#### Derivation for A19:

Choose rule_52: A19 -> A7 13. Use A7 rule_16: A7 -> 1. So yields "1 13". Good.

#### Derivation for A2:

Rule_2 modified: "A2 -> A7 12". Use A7 rule_16 (or any) -> 1. So yields "1 12". Good.

#### Derivation for A20:

Rule_55: A20 -> A7 14. Use A7 rule_16 -> 1. So yields "1 14".

Thus concatenated full sequence:

A23: 7 8 6 1 18 1
A3: 19 10 5 7 2 6 9
A21: 16
A22: 15 4 11 7 8 6 3 17
A19: 1 13
A2: 1 12
A20: 1 14

Combine:

7 8 6 1 18 1 19 10 5 7 2 6 9 16 15 4 11 7 8 6 3 17 1 13 1 12 1 14

Matches target exactly. Great.

Thus the CFG can produce the target sequence with 7 insertions: in rules 2, 7, 15, 17, 25 (two insertions), 56. Wait also need to check we inserted "6" before 1 in rule_56. So the modifications are:

- rule_2: add "12"
- rule_7: add "19"
- rule_15: add "9"
- rule_17: add "15"
- rule_25: add "7" after 11; add "3" after 6
- rule_56: add "6"

Count = 1+1+1+1+2+1 = 7 insertions.

Now consider if any other existence of rule_25 usage in other contexts may cause unintended produce else. But that's okay as we just inserted terminals; it may generate extra tokens elsewhere if A9 is used elsewhere, but not a problem as our derivation does not require those.

Check usage of A9 elsewhere: It appears also in A7->4 A9 (rule_17), which we used for A22's A7 (modified). That's fine. Other uses: A8->2 4 A9 3 (rule_21) etc not used.

Thus modifications may also affect possible other derivations but that's fine.

Now we need to output the new production rules for only those modified.

Thus we must write:

Updated production rules:

rule_2: A2 -> A7 12
rule_7: A3 -> 19 10 A6
rule_15: A6 -> 5 7 A5 6 9
rule_17: A7 -> 15 4 A9
rule_25: A9 -> 11 7 A8 6 3
rule_56: A21 -> 6 1

Now double-check the format: "rule_<n>:" should be same numbering. Provide the RHS with spaces between symbols. It's okay to add inserted terminals positions.

Make sure to include all changed production rules; so "rule_2:", "rule_7:", "rule_15:", "rule_17:", "rule_25:", "rule_56:".

Thus final answer: the six lines.

Potential note: for rule_2, original had just "A7". After insertion "12". Should it be "A7 12" with terminal after A7? Yes.

Check rule_9: There's also rule_56 (modified). No other modifications needed.

Thus final.Updated production rules:
rule_2: A2 -> A7 12
rule_7: A3 -> 19 10 A6
rule_15: A6 -> 5 7 A5 6 9
rule_17: A7 -> 15 4 A9
rule_25: A9 -> 11 7 A8 6 3
rule_56: A21 -> 6 1