We need to understand the problem. We have a context free grammar with many nonterminals. Some production rules have terminal symbols removed (i.e., maybe missing). We need to attempt to repair the CFG to generate the given target sequence, by inserting terminal symbols into the RHS of existing production rules (anywhere). We cannot delete or reorder symbols, nor add new rules. Must minimize total number of inserted terminals across all rules.

Goal: Given the grammar and target sequence, determine modifications (insertions) needed to allow generation of that specific sequence.

We need to find if the grammar can already generate the target string. If not, find minimal insertions to make it possible.

Given constraints: terminals are integers from 1 to 31. The target sequence length: let's count: The target sequence is:

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

Let's count: I'll tally indexes:

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

So it has 41 terminals.

We need a derivation from start A1.

Let’s check the tree: A1 -> A53 A45 A29 A16 A2

Thus derived sequence is concatenation of whatever A53, then A45, then A29, then A16, then A2. So target must be split accordingly.

Goal: The target sequence (41 terminals) should be partitionable into 5 parts: part1 = A53, part2 = A45, part3 = A29, part4 = A16, part5 = A2.

Thus we can assign subsegments correspondingly.

Goal is to find possible derivations for each nonterminal ensuring the concatenated yields the target sequence.

Knowing that some rules have multiple alternatives; we can select whichever fits.

Implementation approach: We can treat this as a parsing problem with partially missing terminals, but we can insert terminals to fill gaps.

Thus we need to find a derivation that matches the target as closely as possible; any missing terminal in a RHS indicates an insertion needed.

Goal: minimize total insertions.

Given the grammar is quite large. Let's analyze each nonterminal's options.

First, A2 is the final part, derived from the target tail: last part of the target sequence (maybe last few terminals).

We see rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A6
rule_4: A2 -> A4 A3 A5
rule_5: A2 -> A12 A3 A11

Thus A2 can generate terminal 1 alone, or more complex sequences. The target ends with "... 1 4 2". The last three tokens in target are 1,4,2. That looks like rule_9: A5 -> 4 2 (i.e., tokens 4,2). So maybe A2 uses rule_4: A2 -> A4 A3 A5. Then A4 -> 1 maybe, A3 -> 1, and A5 -> 4 2 yields sequence: 1 (A4), 1 (A3), 4 2 (A5) = 1 1 4 2. But target end is "1 4 2"? Actually it's "... 1 4 2". There's one 1 before the 4 2. If we can have A4 produce nothing? But A4 rules: A4 -> 1, A4 -> 3. So A4 yields either 1 or 3; cannot produce empty. So A4 yields 1 (or 3). A3 yields 1. So sequence would be "1 1 4 2". target tail: "1 4 2". That's missing one 1 (the second). So maybe we need to insert a terminal into some rule to drop one 1? Actually we cannot delete; only insert. So we cannot get rid of extra 1. Unless we choose alternatives different.

Alternatively, rule_2: A2 -> 1 yields just a 1, but target tail is "1 4 2". Not match; we might need to insert extra terminals after? Might not be enough.

Rule_3: A2 -> 1 A3 A6. A3 -> 1, A6 -> A7 2; A7 -> A4 4; A4 -> 1 or 3. So using A4 -> 1 yields A7 = 1 4, then A6 = (1 4) 2 -> sequence "1 4 2". So A2 production would be "1" (from the leading 1) + then A3 (1) + A6 (1 4 2) => "1 1 1 4 2". That's too many ones.

If we choose A4->3, then A7 yields 3 4 and A6 yields 3 4 2, so overall: "1 A3 (1) A6 (3 4 2)" gives "1 1 3 4 2". Not match.

Alternatively, could we insert extra terminals to adjust?

But perhaps target's final segment includes maybe A2 -> 1 => produce the last 1 before the 4 2? Let's check suffix: The last few tokens: "... 1 5 8 3 1 4 2". So after 1 5 8 3 1, we have 4 2. That 1 before 4 2 is part of A45? Actually we need to find where A45 ends and A2 starts.

Better restructure: Let's try to parse A53, A45, A29, A16, A2 across target.

We'll need to determine a possible partition.

Given A2 likely accounts for the very last part, maybe just "1"? Or "1 4 2"? Let's consider possible partitions.

Define t = target tokens indexed 1..41.

Goal: Divide into 5 parts: p1 (1..i1), p2 (i1+1..i2), p3 (i2+1..i3), p4 (i3+1..i4), p5 (i4+1..41). i0=0, i5=41.

Now find assignments for each nonterminal derived strings.

We need to find derivations for each part using given RHS of rules (with possible insertions). Because insertion can happen anywhere in any RHS; we can augment any rule with additional terminals.

But to minimize insertions, likely best to find derivations that match existing target tokens as much as possible.

Thus the problem is akin to find a parse tree that matches target, where missing terminals are inserted.

Strategy: Build a parsing table with grammar annotated with possible strings, but the target is length 41; many nonterminals produce various strings.

Given number of nonterminals is high, but many produce short strings with numbers up to 31. This looks reminiscent of some mapping of numbers to produce the sequence list.

We need to find a plausible partition.

Another perspective: The target sequence includes many numbers that appear in productions: e.g., 7 appears early second token (the first token is 7). Looking at grammar, we have rule_16: A10 -> 7. So A10 can produce "7". A10 appears in rule_34: A33 -> A10. Also rule_61: A36 -> A10 25. Also rule_33: A10 appears elsewhere.

We need to see if the sequence 7 occurs as part of A53 or A45 or A29 or A16 or A2.

Let's locate possible generating nonterminals for initial token "7". At root A1 -> A53 A45 A29 A16 A2. So first token is from A53 (its derivation). So the first token of target is 7, so A53 must derive a string starting with 7.

Looking at rules for A53:

rule_90: A53 -> 1
rule_91: A53 -> A24 A54 30
rule_92: A53 -> A28 A54 30
rule_93: A53 -> A46 A54 A56
rule_94: A53 -> A57 A54 30
rule_95: A53 -> A60 A54 30

Thus cannot directly produce 7. Perhaps we can insert terminal 7 into some RHS. However maybe we can choose one of the alternatives where first token could be 7 after insertions.

But maybe we can use rule_90: A53 -> 1, but target starts with 7, so we would need to insert 7 before the 1? According to insertion rule, you can insert terminals anywhere in RHS. For rule_90, RHS is "1". We can insert "7 " before, after, or within (but there is only a single terminal, we could insert before, making RHS: 7 1). That would give A53 -> 7 1. But then target's first two tokens are 7 1, which matches? Yes, target starts with 7 1. Good. So one insertion (7) could be added to rule_90 to produce "7 1". That's a plausible approach.

Alternatively, we can insert other tokens elsewhere via other rules. The minimal insertion would be best; adding one terminal to rule_90 yields A53 -> 7 1. That matches the first two tokens.

Now we need to map the rest.

After A53 produces "7 1", the next token from A45 begins at position 3 of target: token is 24.

So A45 must produce a string that begins with 24, and continuing.

Let’s examine the choices for A45:

rule_76: A45 -> 1
rule_77: A45 -> 1 A46 A48 28
rule_78: A45 -> A28 A46 29
rule_79: A45 -> A49 A46 29
rule_80: A45 -> A51 A46 28

These are all the rules for A45. So to generate "24", we could use A45 -> 1 ... but need to match 24 rather than 1. Maybe we need to insert 24 before or after 1. For rule_76, A45 -> 1. To produce 24 as first token, we could insert 24 before 1, making RHS "24 1". That yields 24 then 1. But target has 24 as third token and then next token is 25 (fourth token). So after "24", we need "25". If we have A45 produce "24 1" then next token is 1, not 25. So that's not matching, unless we also insert 25 between? Wait we can insert multiple terminals. Also maybe more complex productions produce longer strings.

Let's examine A45 -> 1 A46 A48 28 (rule_77). Here the RHS yields: "1 <A46> <A48> 28". So 1 then whatever A46, A48 produce, then 28. This yields a string ending with terminal 28. In target, after starting at token 3 (24) we have 25 then 19 ... but maybe A45 yields "24 25 ..." but note A45's production doesn't have 24 or 25 directly except maybe via A46 or A48 expansions.

Alternatively, A45 -> A28 A46 29 (rule_78). Here we have A28, then A46, then 29. A28 can produce 1, 18, 19 via its rules. So not 24.

A45 -> A49 A46 29 (rule_79). A49 produces? rule_86: A49 -> A50 18. So A49 yields something ending with 18 (maybe preceding). Not 24.

A45 -> A51 A46 28 (rule_80). A51 -> A52, rule_89: A52 -> A25, rule_42: A25 -> 1. So A51 yields 1. So rule_80 yields "1 (from A51) <A46> 28". So starting with 1, not 24.

Thus none produce 24 directly except rule_76 if we insert. However we might also produce 24 as part of A46 or A48 expansions.

Let’s see A46: rules: rule_81: A46 -> 1, rule_82: A46 -> 1 A47. A47 -> A24, and A24: rule_41: A24 -> 1. So A46 can produce 1, or 1 A47 => 1 A24 => 1 1 => "1 1". So A46 does not produce 24.

A48: rule_84: A48 -> A24, which yields 1. Also rule_85: A48 -> A28 29 => yields (A28) 29. So maybe A28 can produce 24? Let's check A28 rules: rule_45: A28 -> 1, rule_46: A28 -> 18, rule_47: A28 -> 19. So not 24.

Thus A45 cannot generate 24 as first token without insertion. But we can insert terminals into any RHS, including into right side of A28 etc. But we should aim minimal.

Better approach: maybe our initial partition is wrong; maybe A53 doesn't produce 7 1 but maybe longer, and A45 start later.

However A1's children order fixed: A53, A45, A29, A16, A2.

Thus the target's first tokens correspond as a prefix for A53's generated string. The prefix can be more than 2 tokens; maybe A53 can produce many: like A53 -> A24 A54 30, etc. Need to examine each option.

Given many nonterminals produce sequences that could include 7 later etc.

But we want minimal insertion modifications.

Given we have one clear missing: the first token is 7, which nowhere appears in A53. However, we also have rule_16: A10 -> 7. A10 appears inside A33, A36, etc.

Thus maybe A53 can produce a longer string that includes 7 later (within A54 expansions) maybe solving this without insertion. Let's explore.

We need to check all possible expansions of A53 to see if could generate "7 1 ..." with no insertion.

Rules for A53:

Rule_90: A53 -> 1
Rule_91: A53 -> A24 A54 30
Rule_92: A53 -> A28 A54 30
Rule_93: A53 -> A46 A54 A56
Rule_94: A53 -> A57 A54 30
Rule_95: A53 -> A60 A54 30

We'll need to compute possible derivations.

First see rule_90: yields "1". Not enough for prefix 7.

But maybe we can insert 7 as above.

Now rule_91: A53 -> A24 A54 30. A24 -> 1. A54 -> either "1" or "A40 26 A55". So A24 A54 30 yields "1" + A54 + "30". So eventually begins with 1, not 7.

Rule_92: A53 -> A28 A54 30. A28 -> could be 1, 18 or 19. So starts with 1 or 18 or 19. Not 7.

Rule_93: A53 -> A46 A54 A56. A46 -> 1 or 1 A47 -> ... yields starting with 1. So not 7.

Rule_94: A53 -> A57 A54 30. Need to explore A57.

A57 -> A58 (rule_103). A58 -> A14 A37 A59 (rule_104). A14 can be 1 or 9. A37 -> 1. A59 -> 24 (rule_105). So A58 yields sequence: (1 or 9) , 1 , 24. So A57 yields that sequence: either 1 1 24, or 9 1 24. Then A54 then a 30. So first token maybe 1 or 9.

Rule_95: A53 -> A60 A54 30. A60 -> A61 19 (rule_106). A61 -> A10 A37 A62 (rule_107). A10 can be 1 or 7. A37 ->1, A62->24. So A61 yields either "1 1 24" or "7 1 24". Then A60 yields that plus 19, i.e., ... plus terminal 19.

Thus A53 can start with either "1" (if A10 uses 1) or "7" (if A10 uses 7). So there is a possibility to produce 7 at start of A53 via rule_95.

Specifically: A53 -> A60 A54 30.
A60 -> A61 19.
A61 -> A10 A37 A62.
A10 -> 7 (via rule_16). Then A37 -> 1 (rule_63). A62 -> 24 (rule_108).
Thus A61 yields "7 1 24". Then A60 yields "7 1 24 19". Then A54 yields something (maybe "1" or "A40 26 A55"). Then we have terminal "30". So A53 can start with 7 1 24 ...

Our target starts with "7 1 24 25...". After 7 1 24, the target has 25. So after "7 1 24", we need "25". Let's see A53's remainder after generating "7 1 24 19" we get "7 1 24 19". However target after "7 1 24" is "25". So the 19 appears later (target token at position 5 = 19). Actually target: 7 1 24 25 19 ... The 19 appears at position 5. In A53 derived using rule_95, after "7 1 24", we have "19" (coming from A60). That's at position 4? Let's see: A53 yields "7 1 24 19...". But target has "7 1 24 25 19". So there's a mismatch: we need "25" before "19". The "25" could come from A54 maybe? Because after A60, there is "A54", then terminal 30. Let's evaluate A54 possible expansions.

A54 rules:

rule_96: A54 -> 1.
rule_97: A54 -> A40 26 A55.

Now potential jump: we need to generate "25" perhaps from A54 -> A40 26 A55, where A40 can be:

rule_68: A40 -> 1.
rule_69: A40 -> 1 A41.

Thus A40 can produce "1" or "1 <A41>".

A41 rules: rule_70: A41 -> A23; rule_71: A41 -> A28 17.

A23: rule_39: A23 -> 1. rule_40: A23 -> 15.

Thus A41 could produce 1 (if A23->1) or 15 (if A23->15) or A28 17.

A28 possibilities: 1,18,19.

So A41 could produce "1" (via A23->1) after preceding 1 from A40? That's like "1 1". Or "1 15". Or "1 1 17". Or "1 18 17". Or "1 19 17". Or maybe just "1"? Wait A41->A23 without any preceding symbol; A40->1 or 1 A41.

If we choose A40->1 and A41->A23->? Actually A54 is A40 26 A55, so we have A40 (maybe 1 or 1 A41), then terminal 26, then A55 (A55 -> A23 or A25). This yields possibly sequences containing 26.

Our target after 7 1 24 25 19 ... but we need 25 after "7 1 24". How can we generate a 25? Maybe from A54 via A40->1 A41, A41->A23->15? That's 1 then 15 then 26 then A55->? But no 25. 25 is present as terminal ID #25 (i.e., 25). There is a rule: A36 -> A10 25 (rule_61). So 25 appears as part of some production but not in A54.

Potentially, A25 -> 1 (rule_42). So not 25.

Terminals in productions: We see target includes 25 as the fourth token. Possibly A45 generates "25" or other nonterminal.

Let's look at all productions for 25 directly: rule_61: A36 -> A10 25. A10 could be 1 or 7. So A36 can generate "1 25" or "7 25". Also rule_55: A32 -> 22 A33. A33 can be A10 (rule_56), so A32 could have "22 A10". And A33 -> A14 23 (rule_57). A14 can be 1 or 9, leading to "1 23" or "9 23". But not 25.

Also rule_77: A45->1 A46 A48 28 includes 28 at the end. Not 25.

Rule_78: A45->A28 A46 29, where A28 can be 1, 18, or 19.

Not obvious.

Thus 25 appears only in rule_61 (A36 -> A10 25) and maybe other places? Let's search: rule_39? Not.

Thus it's likely that our derivation for A45 or A29 or A16 includes A36 to produce 25.

Nevertheless, we need to parse whole.

Better approach: Use systematic parsing. Because trying to hand parse huge grammar may be complex. However, we can try an exhaustive approach with backtracking manually.

Observations: The target contains many distinct numbers. The grammar seems to be a constructed example whereby each number appears in some production and the target is possibly the concatenation of many bits.

Goal: Insert minimal terminals. Likely only need to insert a few to cover numbers that can't be matched.

Potentially the original grammar might have been enough to produce the exact target sequence, but some terminal symbols were removed. Need to determine which ones.

The question: "the CFG production rules have had some terminal symbols removed which may now render it impossible to produce the target sequence. Please attempt to repair the CFG so that the sequence can be produced. The only operations which can be made to repair the CFG are the insertion of terminal symbols anywhere in existing production rules. ... minimize total inserts."

Thus there is a correct minimal set of insertions that would reconstitute original grammar precisely to produce target. Likely it's small.

Thus we need to think about missing terminals in each rule: maybe each rule originally had some terminals after nonterms but they were removed.

Thus to find modifications, we could treat each production as nearly complete, but missing some tokens.

We could try to find a derivation that exactly matches the target when we assume each rule used is as given, but where we need to insert missing terminals along the way.

We need to find a parse tree with minimal "missing" terminals; each missing terminal corresponds to insert a terminal in the RHS where a rule's RHS is missing a terminal that should be there.

Thus it's like tree edit: we can insert terminals into RHS.

Simplify: Since they want minimal inserts, we can seek a parse where each nonterminal's expansion has the form needed; any mismatched terminal not present requires insertion.

Thus we should match the target string as a concatenation of the expansions (including both present terminals and inserted ones). Inserted terminals can be anywhere.

Thus to compute minimal insertions, one can treat each rule expansion as a sequence of alternating terminals and nonterminals. We can match the target segment to that expansion, inserting missing terminals where needed.

Thus for each rule used, the cost (number of insertions) is the number of positions where the rule's RHS does not have a terminal that we need.

Thus we need to find a derivation with minimal total cost.

We need to do a kind of parsing with weighted costs.

We can solve this by dynamic programming: each nonterminal can derive any substring of the target with minimal insert cost.

Because the grammar is not huge (62 nonterminals, productions about 108 rules). Could compute DP for substrings indices.

However, as a reasoning exercise, we can approximate manually.

But perhaps easier: one can search for a plausible derivation that matches exactly with zero insertions if possible. Given the target includes all numbers, there may be an exact derivation using existing productions with maybe one or two missing.

Alternatively, maybe there is a simple solution: Insert "7" before rule_90 as we saw, and also insert "25" after rule_41 maybe. But need to examine.

Given A53's derivation using rule_95 produces 7 early. So maybe we can use that alternative.

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

We need to segment this into 5 parts as previously.

Let's attempt to fit a parse using existing productions.

Segment candidate: A53 corresponds to first part of target. The grammar options for A53 start with 1, 1/18/19 etc, or sequences containing 7. Notably A53 -> A60 A54 30 yields a fairly long prefix.

Goal: see if we can map target's prefix (maybe up to some point) to A53.

Start with rule_95 (A53 -> A60 A54 30). That yields: <A60> <A54> 30.

Target ends with ... 30 appears at position 14 (the 14th token is 30). Yes, the target token 14 is 30. So maybe A53's 30 matches that. If A53 ends with 30, then A54 provides stuff leading up to that.

Thus A53 may generate tokens from position 1 up to including the token before 30 (position 13 = 31). Indeed target token positions:

1:7
2:1
3:24
4:25
5:19
6:1
7:16
8:1
9:17
10:26
11:15
12:27
13:31
14:30

So perhaps A53 covers tokens 1..14 as "7 1 24 25 19 1 16 1 17 26 15 27 31 30". This matches: The substring starts with 7 1 24, includes 25, etc. That's a possible A53.

Thus we suspect A53 generates exactly the first 14 tokens (ending with 30). Then A45 starts at token 15 (11). Then rest accordingly.

Let's test if A53 can generate that exactly using existing productions without insertion. Since 30 is at the end of A53 production (via rule_95). So that's plausible.

Now we need to find derivation of A53 that yields tokens up to 30.

We have A53 -> A60 A54 30.

Thus we need to produce tokens 1..13 (excluding final 30) via A60 and A54.

That is:

A60 then A54 then terminal 30.

Thus we need production of A60 yields tokens 1..some index, then A54 yields some tokens, then terminal 30.

Given A60 -> A61 19 (rule_106). So after A60, there will be a terminal 19. Indeed target token pos5 is 19. Good. So A60 yields everything before that 19 (including 7 1 24 25) and then 19. Actually A60 yields A61 (which yields something) then 19. So token ordering: [A61's output] likely includes 7, 1, 24, 25 (maybe) and then after that A61's content ends; then token 19 from rule_106.

Thus far matches target: tokens 1..5 = 7 1 24 25 19. So A61 must derive "7 1 24 25". Fill that.

A61 -> A10 A37 A62 (rule_107). So A10, then A37, then A62. A37 -> 1 (rule_63). A62 -> 24 (rule_108). So A61 yields: [A10] 1 24.

We need A61 to generate "7 1 24 25"? No, A61 yields something that ends right before the 19, but A61 yields exactly A10 1 24. That is three tokens: A10, then 1, then 24. Then after A61, rule_106 adds 19. So total would be [A10] 1 24 19.

Thus far: A61 -> A10 A37 A62 yields tokens: (A10) then 1 then 24. So sequence "A10 1 24". Then A60 adds a terminal 19, making "A10 1 24 19". Target's first five tokens are "7 1 24 25 19". So needs "A10" produce "7 1 24 25"? But that seems too many tokens.

Consider that A10 can produce “7” (via rule_16) or “1” (rule_15). But alone A10 produces a single token. So A10 cannot produce "7 1 24 25". It just produces either 7 or 1. So A61 yields "7 1 24" if A10 selects 7. Then A60 adds 19 => "7 1 24 19". This would be tokens "7 1 24 19". But target is "7 1 24 25 19". There is an extra 25 before the 19. So A54 likely is responsible for inserting 25 between the 24 and 19? Wait sequence is A60 A54 30; Actually A60 produces all up to its own terminal 19 then A54 then 30. Wait we must be careful: A53 -> A60 A54 30. So after A60 (which ends with 19), we then have A54, then 30. So A60 yields tokens up through the 19, then A54 yields tokens between 19 and 30. So target portion from after A60's 19 up to before the final 30 should be produced by A54.

Thus tokens after 19 (position 5) up to before 30 (position 13) should be from A54. Those tokens are: 1,16,1,17,26,15,27,31. Let's list:

Positions:
6:1
7:16
8:1
9:17
10:26
11:15
12:27
13:31

Thus A54 must derive substring "1 16 1 17 26 15 27 31". Does any production for A54 produce that? A54 has rules:

rule_96: A54 -> 1
rule_97: A54 -> A40 26 A55.

Thus only two alternatives. The first yields just "1". Not enough. The second yields "A40" then terminal 26 then A55.

Thus we need to produce "1 16 1 17 26 15 27 31". Notice there is a terminal 26 already in the RHS after A40. Target's substring contains a 26 at position 10 (the 5th token of the block). Indeed after preceding part "1 16 1 17", we have "26". This could align with the 26 from rule_97. Then preceding tokens (1 16 1 17) would come from A40? Let's examine A40's possible expansions.

A40 rules:
rule_68: A40 -> 1
rule_69: A40 -> 1 A41

Thus A40 yields either just "1" or "1 <A41>". That gives a leading 1 and then maybe more.

Thus A40 can produce "1 <A41>" where A41 begins after the initial '1'.

Thus the segment from A40 needs to produce tokens "1 16 1 17"? Let's see: Starting at A40, we could have "1" (rule_68) then we need to produce the rest "16 1 17". But rule_68 only gives "1". So not enough. If we choose rule_69: A40 -> 1 A41. Then we have "1" then A41's derivation. So after that "1", we need "16 1 17". So A41 must generate "16 1 17". But per rules:

A41 -> A23 (rule_70) or -> A28 17 (rule_71).

Option A41 -> A23 yields just A23, which can be "1" (rule_39) or "15" (rule_40). That yields a single token. Not enough for "16 1 17".

Option A41 -> A28 17 yields A28 then terminal 17. So A41 yields something + 17. So after initial '1' from A40, we need to get "16 1" from A28? Because A41 would add something from A28 before the terminal 17. So A28 must derive "16 1". But A28 can only produce "1", "18", or "19". So "16" not possible. So we cannot produce "16" via A28.

Thus the needed terminals "16" and "17" must come from somewhere else: perhaps they are part of A41's expansions using A23? No, 16 not present.

Where else is 16 present? Let's search the productions for 16. I recall rule_24: A16 -> 1 A17 A22 10. Also rule_25: A16 -> A18 A17. Also A24 -> maybe 16? Not. Also rule_61: A36 -> A10 25. Not include 16. Actually 16 appears as terminal ID "16". Let's search: In the production list, there is rule_24 (in A16) includes terminal "10" not 16. The integer "16" appears as a target token but not obviously as a terminal in any production. Let's scan entire list:

Terminals in productions: It includes numbers 1,3,4,2,6,7,8,9,5,8? Actually 8 appears as terminal in rule_22: A15 -> 5 8. 3 appears in rule_8: A4 -> 3. 10 appears in multiple rules. 11 appears in rule_32: A18 -> 11; rule_33: A19->1; Actually "11" appears as a terminal. 12 appears in rule_50? Actually there is rule_50: A29 -> A9 A31 A30 (30 includes 21 20). No 12. But "12" appears maybe in rule_?? Not sure.

But 16 appears only in target; no rule has literal terminal 16. Let's confirm: Check list for any "16". rule_24: A16 -> 1 A17 A22 10 (not terminal 16). rule_25: A16 -> A18 A17 (no 16). rule_26: A16 -> A19 A17 10 (no). rule_27: A16 -> A20 A17 10. rule_28: A16 -> A26 A17 14 10. So terminal 16 doesn't appear in any rule (likely missing). That explains why target includes terminal 16 but grammar doesn't have 16 anywhere. So we must insert terminal 16 somewhere in productions. The only location where 16 could be inserted is perhaps within a rule that originally included 16 but got removed.

Which rule could originally have had 16? Perhaps A40's production (A40 -> 1), but unlikely. More plausible is rule_14: A9 -> 6 (but could have been 6 and some other? Not 16). Actually, A17 is a nonterminal producing 1 or 1 A21, but not 16.

Terminal 16 appears in target at position 7 (7:16). After the initial sequence "7 1 24 25 19 1" we have "16". Indeed position 7 is 16. So after A60 (which gave up to 19) and A54 (which maybe gave up to token before 30), we need 16 within A54's output. However A54's production does not include 16.

Thus likely we need to modify a rule to include terminal 16.

We need to find a rule that could hold 16 as part of the sequence "1 16 1 17". Perhaps it's A40 again. Or maybe A41 includes 16. There is rule_71: A41 -> A28 17. Could have originally been "A28 16 17"? But the rule currently has 17 only. But we need both 16 and 17. So maybe we need to insert 16 before 17 in rule_71.

Thus modify rule_71 to be: A41 -> A28 16 17, i.e., insert the terminal 16 before 17. That would allow deriving "A28 16 17". Then A41 can produce 16 and 17, which matches the needed tokens.

Let's check: The needed substring after the initial "1" from A40 is "16 1 17". Actually we want "16 1 17": token order is "16 then 1 then 17". Where would 1 come from? Possibly from A28 (because A28 can generate 1). So A28 = 1, then we need 16, then 17? No, order is A28 (1), then maybe terminal 16, then terminal 17. That's "1 16 17". But our substring needed is "16 1 17". That's different order.

Our withheld substring is "1 16 1 17"? Actually from earlier we said A54->A40 26 A55. Let's line up more systematically.

We determined the target for A54 is "1 16 1 17 26 15 27 31". The ordering: tokens 6..13: 1,16,1,17,26,15,27,31.

Thus after A60 (which gave up to token 5=19), we have token 6 = 1. That aligns with A54’s first token. The token after that is 16, then 1, then 17, then 26, then 15, then 27, then 31.

Thus we need to produce sequence from A54: "1 16 1 17 26 15 27 31".

But A54's production "A40 26 A55" yields: prefix from A40, then terminal 26, then from A55.

Thus the prefix from A40 must be "1 16 1 17". Then after 26, A55 must generate "15 27 31". Let's verify.

A55 -> A23 (rule_98) or A25 (rule_99). A23 -> 1 or 15. A25 -> 1. So A55 can generate "1" or "15" or "1". But we need to generate "15 27 31": three tokens. That seems too many.

Thus A55 alone not enough; perhaps A55 generates "15" and then we need subsequent symbols for 27 and 31 from later expansions? Wait where do 27 and 31 come from? Let's see other productions that produce 27 and 31.

Terminal 27 appears in rule_41? Actually rule_41: A24 -> 1 only. So A24 cannot produce 27. However, note rule_87: A50 -> A18 1 12. This yields 12. Also rule_90 etc.

Search for "27". The only place I see is target token 27. Check productions: There might be rule_31: A18->1? Actually no. Search through list for terminal 27: I see in rule_... Might need to locate explicitly. Let's scan the list:

- rule_24: A16 -> 1 A17 A22 10 (includes terminal 10)
- rule_25: A16 -> A18 A17 (no)
- rule_26: A16 -> A19 A17 10
- rule_27: A16 -> A20 A17 10
- rule_28: A16 -> A26 A17 14 10 (contains 14)
- rule_40: A23 -> 15 (that's 15)
- rule_41: A24 -> 1
- rule_42: A25 -> 1
- rule_45: A28 -> 1
- rule_46: A28 -> 18
- rule_47: A28 -> 19
- rule_48: A29 -> 1
- rule_50: A29 -> A9 A31 A30 (A30 -> 21 20)
- rule_54: A31 -> A34 A32 (no directly)
- rule_55: A32 -> 22 A33
- rule_56: A33 -> A10
- rule_57: A33 -> A14 23
- rule_61: A36 -> A10 25
- rule_62: A36 -> A14
- rule_70: A41 -> A23 (no)
- rule_71: A41 -> A28 17
- rule_73: A43 -> A44 6 (terminal 6)
- rule_74: A44 -> A23 A40 26 27 (terminals 26 27)
- rule_75: A44 -> A23 A40 27 (27 only)
- rule_78: A45 -> A28 A46 29 (terminal 29)
- rule_79: A45 -> A49 A46 29
- rule_80: A45 -> A51 A46 28
- rule_81: A46 -> 1
- rule_82: A46 -> 1 A47
- rule_87: A50 -> A18 1 12 (terminals 1,12)
- rule_91: A53 -> A24 A54 30 (30)
- rule_92: A53 -> A28 A54 30 (30)
- rule_93: A53 -> A46 A54 A56 (no direct)
- rule_94: A53 -> A57 A54 30
- rule_95: A53 -> A60 A54 30

Thus we have 27 present as a terminal in rule_74 and rule_75 (A44 productions). Also in rule_74's sequence: A23 A40 26 27. Also on target, after 15 (token 11) we have token 12 = 27. So the substring "15 27" indeed appears in target. This suggests we may need to involve A44 to produce "15 27". Indeed A44 -> A23 A40 26 27 (rule_74) yields "A23" (maybe 15), "A40" (maybe 1?), terminal 26, terminal 27. However our target substring after token 10 = 26 is token 11 = 15, token 12 = 27. Actually "15 27" are after 26 not preceding it. Our substring after 26 is "15 27". But rule_74 yields "A23 A40 26 27". That's reversed: after A23 A40, then 26 then 27. The order 26 then 27 is okay, but 15 appears before 26. Our target has "15 27" after 26, but the rule places 15 (if A23->15) before 26. Not matching.

But maybe there is A44 -> A23 A40 27 (rule_75) which yields A23 then A40 then 27. That would give something like (maybe 15) then (some A40) then 27. But still the 27 appears after A40, but there is no 26.

Our target in that region is "15 27". Not 27 after some extra. So we need to produce exactly "15 27". Check if there is any rule that yields "15 27". Not directly.

However, perhaps A55 -> A23 (if A23 -> 15) yields 15, and later some other rule yields 27 from some nonterminal.

Let's search for nonterminals that can produce 27. A44's 27 is a terminal. Also maybe A39? No. A45? No. A50? No. So only way to produce 27 is via A44 (rule_74 or rule_75). Also A51? No. So we need to include A44 somewhere.

But after token 13 = 31, the next token (pos14) is 30, which is A53's termination.

Thus for A53, after A54 and before 30 we include token 30. So A53's A54 outputs "1 16 1 17 26 15 27 31". After 31 then the rule ends with 30, making 30.

Now we must see if A54 can produce "1 16 1 17 26 15 27 31". For the suffix "15 27 31", we see that 27 is maybe from A44 substructure.

But A54's A55 after the 26 produces something. A55's alternatives: A23 (produces 1 or 15) or A25 (produces 1). So from A55 alone we cannot output three tokens "15 27 31". So we must somehow incorporate A44 output inside A55? No, A55 cannot interface with A44 as is. So maybe we need to consider that we actually mis-assigned segment to A54; perhaps some of those tokens are part of A53 before A54, not after. Actually A53's RHS after A60 is A54 30. So A54's output must be directly before 30. So tokens before the final 30 (position 13 =31) may be part of A54. Actually the token before 30 is 31. So 31 must be part of A54's output (since after A54 we have terminal 30). So A54 must produce tokens up to 31. It's plausible.

Thus A54 must produce 1 16 1 17 26 15 27 31.

Given the apparent difficulty, maybe we can insert missing terminals into A54's production to allow more. Since we can insert terminals anywhere, maybe we can directly insert the tokens needed before or after sub-nonterminals. For instance, modify A54's RHS: currently "A40 26 A55". We could insert terminals 16, 1, 17 (what needed) and also 27 and 31.

Specifically, we could directly insert "16 1 17 31" after A55? Or we could replace A55 with something bigger? But we are not allowed to delete or reorder existing symbols; we can only add terminals. So base order must be: (any inserted terminals before ) A40 (maybe with inserted terminals around) 26 (maybe additional) A55 (maybe with inserted after) and then after A55, no more RHS because the rule ends. However after A55 ends, we then have the terminal 30 from A53.

Thus we can insert terminals before or after each existing symbol. So we can insert "16 1 17 31" after A55, before the 30 (but between A55 and 30 there is already nothing; but we may consider inserting after A55 for it to be before the 30. That addresses tokens beyond A55.

Thus we could have A54 produce "A40 26 A55" but we can also insert extra terminals after each symbol. So we can put needed terminals after A55 (like 31) and also before 26 perhaps. Let's try to construct a mapping.

Goal: Based on A54 RHS, need to produce sequence:

[maybe inserted] A40 [maybe inserted] 26 [maybe inserted] A55 [maybe inserted].

We need to match "1 16 1 17 26 15 27 31". Recognize that there is exactly one terminal 26 which matches the 26 in target at position 10. Good.

Thus A40 and A55 must produce the rest.

Let's consider A40 -> 1 A41 (rule_69). This yields "1" then whatever A41 yields. So A40 can produce "1" plus something.

Thus we could map:

- The first "1" in target (position 6) could be provided by A40's leading "1".
- Then we need "16 1 17" before the "26". So we can insert those after the leading 1, before the 26, maybe directly in the A40 production or after it. However we can only insert terminals into A40's RHS (not into its expansion after we expand). But A40's RHS is "1 A41". After expanding A40, we have "1" then whatever A41 yields. In between the "1" and the expansion of A41 we cannot insert terminals unless we modify RHS to include them: e.g., "1 16 1 17 A41". But we cannot reorder; we can insert new terminals anywhere in RHS. So for A40 rule, we could change to: "1 16 1 17 A41". Alternatively, we could also insert after A41.

But the 26 terminal comes after A40 in A54's RHS: A40 (some terminals) then terminal 26, then A55. So A41's expansion will be before 26 (since it's part of A40). So we need "16 1 17" to appear before the "26". So this can be inserted into A40's RHS after the "1". Good.

Thus modify rule_69: currently A40 -> 1 A41. Insert terminals 16 and 1 and 17 before A41. So rule becomes: A40 -> 1 16 1 17 A41. Or we could place them after A41 but then they would be after 26, which is not correct.

Thus we propose to insert three terminals: 16, 1, 17 into rule_69 (or rule_68). Possibly we can also use rule_68 (A40 -> 1) and add the missing tokens after 1 within rule_68: "1 16 1 17". But then A40 would not have A41 to generate further. However A40 must have a nonterminal? Not necessary if we choose rule_68. Using rule_68 we could produce "1 16 1 17" from A40 (by inserting in rule_68). Then the following token in A54's RHS after A40 is 26, which would place those tokens before 26 correctly. That may be simpler: modify rule_68 to be "1 16 1 17". This inserts 3 terminals (16,1,17). Then after A40, we have 26, which matches target token 10.

Thus we don't need A41 at all for this part. So we can keep rule_69 unchanged. But we need to choose which rule for A40 to use in A54's production; rule_68 yields just that, rule_69 yields 1 plus A41; but we can still choose rule_68.

Thus A40 expands to "1 16 1 17". Then A54 yields "1 16 1 17 26 A55". So far we have matched tokens 6..10 ("1,16,1,17,26").

Now we need to match tokens 11..13: 15,27,31 with A55 and maybe inserted terminals after A55 before 30.

Recall target after 26 is "15 27 31". So A55 must produce something that can correspond to these, maybe with insertions.

A55 alternatives: A23 or A25. Let's examine each.

Option A55 -> A23 (rule_98). A23 can be 1 (rule_39) or 15 (rule_40). So we could produce "15" if using A23->15. That matches the first token (15). Good. Then after A55, we have the final 31 before the final 30.

Thus we need to produce "27 31". Since A55 after producing 15 cannot produce more tokens, we could insert terminals after A55 (within A54's RHS after A55) before the final 30: we can insert "27 31" as two terminals. That would match tokens 12 and 13. So modify rule_97 (A54 -> A40 26 A55) to also have inserted terminals after A55: e.g., "A40 26 A55 27 31". Wait we need to check that after A55 we have the next token in the generated string, which proceeds to terminal 30 (from A53). So by inserting "27 31" after A55 in the RHS of rule_97, we cause A54 to generate "A40 26 A55 27 31". Then A53's final 30 yields the last token. This matches target exactly: after A55 (which yields 15), we produce 27, then 31, then 30.

Thus we add two insertions to rule_97.

Now check we have matched the entire A53 substring: For A53 using rule_95: A53 -> A60 A54 30.

A60 produces "A61 19". A61: A10 A37 A62 yields "7 1 24". Wait we need "7 1 24 25" before the "19". Actually we need "7 1 24 25 19". Where does 25 come from? Currently A60's expansion goes: A61 (7 1 24) then 19 -> yields 7 1 24 19, lacking 25. So we need to insert a terminal 25 somewhere between the 24 and 19. That could be inserted in the RHS of A61 or A60. Let's examine.

A61 -> A10 A37 A62 yields 7 (from A10) 1 (A37) 24 (A62). There's an order "7 1 24". We need to insert "25" after that 24, before the 19 from A60.

We could modify rule_106's RHS: A60 -> A61 19. Insert 25 before the 19, or maybe after A61's expansion by modifying rule_106 to "A61 25 19". That adds one insertion of terminal 25. That would produce "7 1 24 25 19". Perfect.

Thus we need to insert terminal 25 into rule_106.

Now we have A60 produce "7 1 24 25 19". Then A54 produce rest and final 30.

Thus A53 adequately matches first 14 tokens with minimal insertions: we inserted 3 terminals to rule_68 (16 1 17), 2 terminals to rule_97 (27 31), and 1 terminal to rule_106 (25). Total 6 insertions.

But we also inserted possibly adjustments for rule_68; is that needed? Actually we could also achieve the same result using rule_69 with A41 inserted? Let's check alternative: rule_69: A40 -> 1 A41. But we previously decided to modify rule_68 because it's simpler. However we need to ensure only one rule per nonterminal may be used; multiple RHS exists; we can choose any. As long as we modify some rules to allow match.

Thus far we have modifications:

- rule_68: A40 -> 1 -> modify to: A40 -> 1 16 1 17 (insert 16 1 17 after 1). That's "1 16 1 17". Insert three terminals.

But note that could be considered as inserting after "1". So exactly 3 insertions.

- rule_97: A54 -> A40 26 A55, modify to "A40 26 A55 27 31". Insert "27 31". 2 insertions.

- rule_106: A60 -> A61 19, modify to "A61 25 19". Insert 25. 1 insertion.

Total 6 insert.

Now next parts: after A53, we have A45 generating the next token(s) starting at position 15 = 11.

Thus after A53 (ending with token 30), the next token (position 15) is 11.

Thus A45 must derive a string starting with 11 and possibly covering subsequent tokens, maybe up to some point.

Target after A53: tokens:

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

Thus we need to parse the remaining tokens with A45, A29, A16, A2 sequence.

Now we need to use grammar for these.

Let's inspect A45 productions.

We have 5 alternatives: rule_76 (A45 -> 1), rule_77 (1 A46 A48 28), rule_78 (A28 A46 29), rule_79 (A49 A46 29), rule_80 (A51 A46 28).

Goal: starting token 11. None produce 11 directly. So we need insertion(s) to get 11.

Potentially, we could also produce 11 within a nonterminal used inside A45. Let's investigate.

Option A45 -> 1 (just 1). Not 11.

Option A45 -> 1 A46 A48 28. RHS includes terminals: 1, 28 at end, plus whatever expansions of A46 and A48. A46 expansions (rule_81: 1; or rule_82: 1 A47). A48 expansions (rule_84: A24, which yields 1; or rule_85: A28 29). So overall A45 can produce a string that starts with 1.

Thus not produce 11 without insertion.

Option A45 -> A28 A46 29. A28 can be 1, 18, 19; A46 again yields 1 ... So starting token maybe 1,18,19 but not 11.

Option A45 -> A49 A46 29. A49 -> A50 18; A50 -> A18 1 12 (that includes terminal 12). So A49 yields sequence from A50 and then 18: A50 yields A18 1 12, A18 could be 1 or 11, per rule_31 or rule_32 (A18->1 or 11). So A49 expands to either (1 1 12) 18 ? Actually A18 can be 1 or 11. So if A18->11, then A50 yields 11 1 12, then A49 adds 18 => yields 11 1 12 18. So A45 option gives A49 A46 29: we get prefix "11 1 12 18" from A49, then A46 yields something (maybe 1) then terminal 29.

Thus A45 using rule_79 could produce a string starting with 11, which matches the target's token 15=11. Let's analyze the sequence.

We need to match target from token 15 onward: "11 1 12 13 18 1 29 28 6 1 22 9 23 21 20 11 1 14 10 1 1 5 8 3 1 4 2"

But note after 11, we have 1 12 13 18 ... Let's see A49 A46 29 path: A49 yields "11 1 12 18". After that, A46 yields e.g., "1" or "1 A47" (i.e., maybe more). Then after that we have terminal "29" from rule_79. So overall start of A45 (using rule_79) yields: "11 1 12 18" + (A46 expansion) + "29". The target after the initial "11" is "1 12 13 18 1 29". Compare: after "11" we have "1 12 13 18 1 29". Our derived sequence is "11 1 12 18 ..." missing a "13" before 18. Also order: we have 18 immediately after 12. Need a 13 before 18.

Thus we may need to insert terminal 13 into A49's expansion, maybe after 12 or before 18. Let's see where 13 could be inserted: A49's RHS is "A50 18". A50 expands to "A18 1 12". That's "A18", then 1, then 12. So we have sequence: [A18],1,12,18. A18 may be 11. So sequence: 11,1,12,18. Need to insert 13 after 12 and before 18. So modify rule_86 (A49 -> A50 18) to insert terminal 13 before the final 18: i.e., "A50 13 18". That would give "11 1 12 13 18". Good.

Thus one insertion in rule_86.

Now after that we have A46's expansion. Currently after 18 we have A46 (which could produce "1" or "1 A47"). The target after 18 is token 20 = 1. So we can have A46 -> 1 (rule_81). That yields just 1. Good.

Now after that in A45, we have terminal "29". The target after that 1 is token 21 = 29. So matches. Thus rule_79 yields A45 => "11 1 12 13 18 1 29". This matches tokens 15-21 exactly. Perfect.

So for A45 we used rule_79 (A49 A46 29) with one insertion (13) in rule_86.

Great.

Thus after A45, the next part is A29 starting at token 22 = 28. Because A45 covers up to token 21 = 29. Indeed token ordering: after 29 (token 21) next token is 28. That's token 22.

Now parse A29.

Production rules for A29:

rule_48: A29 -> 1
rule_49: A29 -> 1 A31 A38
rule_50: A29 -> A9 A31 A30
rule_51: A29 -> A43 A31 A42

Thus to start with terminal 28, we need insertion because none have leading 28. But perhaps via rule_49: A29 -> 1 A31 A38; we could insert terminals before 1? We can insert before "1". However the token we need as first is 28; so we could insert "28" before the 1. That yields "28 1 ..." However target's token 22 is 28, token 23 is 6?

Check target tokens from 22 onward: token 22=28, token23=6, token24=1, token25=22, token26=9, token27=23, token28=21, token29=20, token30=11, token31=1, token32=14, token33=10, token34=1, token35=1, token36=5, token37=8, token38=3, token39=1, token40=4, token41=2.

Thus we need A29 to generate "28 6 1 22 9 23 21 20 11 ...". Potentially this is a long token sequence.

Let's examine possible expansions of A29.

Option rule_48: just 1; not appropriate.

Option rule_49: "1 A31 A38". A31 expands maybe to many tokens. A38 expands maybe to something else.

Option rule_50: "A9 A31 A30". A9 can be 1 or 6 (rule_13 and rule_14). Good: A9 can produce 6, which matches token 23 perhaps. A31 can produce many things, but let's examine.

Option rule_51: "A43 A31 A42". A43 leads to A44 6; includes 6 after A44.

So we look at target after 28: token 23 is 6. So we might want to produce "6" from A9 or from A43->A44 6.

Thus we could either use rule_50: start with A9, which could produce 6 (using rule_14). However A29's RHS for rule_50 begins with A9, not preceded by 28. So we need to generate 28 before A9. That could be done by insertion before A9 (i.e., insert terminal 28 ahead of A9). So rule_50 could be patched as "28 A9 A31 A30" or we could add "28" before A9.

Alternatively, we could consider rule_51: A43 A31 A42, where A43 -> A44 6, thus begins with A44 then 6. So rule_51 would produce something that eventually starts with maybe a terminal from A44 then 6. But we need leading 28 then 6. So we could insert 28 before A43 or inside A44.

The grammar also has A44 productions that include terminal 27 and 26.

Check rule_74: A44 -> A23 A40 26 27. This yields sequence: (A23), (A40), 26, 27. A23 can be 1 or 15; A40 can produce 1 and maybe more. So A44 may produce "1 1 26 27" or "15 1 26 27" etc. Not 28.

But 28 might be generated elsewhere. Look at productions for 28: appears in rule_77: A45 -> 1 A46 A48 28; in rule_80: A45 -> A51 A46 28; also direct terminal 28 appears as itself only there. Also rule_95: A53 -> A60 A54 30; no. Rule_95 no 28. Actually rule_77: A45 ends with 28. That's separate.

Also rule_90? Not.

Thus to get token 28 as part of A29, we must insert it (since it's not produced elsewhere). However perhaps 28 appears as part of A40 expansions? A40 expansions: "1" and "1 A41". No.

Thus we need insertion of 28.

Now, after that, the target's next token sequence after 28 and 6 is: 1 22 9 23 21 20 11 1 14 10 1 1 5 8 3 1 4 2.

This appears to be derived from A31 and perhaps A38, A30, etc.

Option rule_49: A29 -> 1 A31 A38. That yields initial 1, not 28, but we could insert 28 before 1. Then after that we have A31 then A38. Let's examine A31 production.

A31 alternatives: rule_53: A31 -> 1. rule_54: A31 -> A34 A32.

Thus A31 can produce 1, or produce the concatenation of A34 and A32.

Probably we need to generate a lot of tokens, so probably we need the longer production.

Let's examine A34 productions:

rule_58: A34 -> 1.
rule_59: A34 -> A37 A35.

Thus A34 can produce 1, or A37 A35. A37 -> 1. A35 -> 24 A36 (rule_60). So A34 -> 1 A35 -> 1 24 A36 (if using rule_59). A36 can be "A10 25" or "A14". So possible strings could be "1 24 (A10 25)" or "1 24 (A14)". Let's expand A36 possibilities: rule_61: A36 -> A10 25 (A10 = 1 or 7). So could be "1 25" or "7 25". Or rule_62: A36 -> A14 (where A14 is 1 or 9). So A36 could be "1" or "9".

Thus A34 via rule_59 yields e.g., "1 24 1 25" or "1 24 7 25", or "1 24 1". Or via A34 -> 1 yields just "1".

Now A32: rule_55: A32 -> 22 A33. So yields "22" then A33. A33 alternatives: rule_56: A33 -> A10 (i.e., 1 or 7). rule_57: A33 -> A14 23 (i.e., 1 or 9, then 23). So A32 yields "22" followed by either "1" or "7" or "1 23" / "9 23". So typical strings: "22 1", "22 7", "22 1 23", "22 9 23".

Thus A31 via the longer production A34 A32 yields sequences like:

Option: A34->1, A32->22 A33 -> => "1 22 ...". Or A34->A37 A35 -> yields "1 24 ...". Combined yields something like "1 24 ... 22 ..." but ordering is not defined: A34 first then A32. So sequence for A31 = [A34 expansion] + [A32 expansion].

Now target after "6 1 22 9 23 21 20 11 ...". Let's see: after "6" we have "1 22 9 23 ...". That matches the pattern: "1" from possible A34 (or just token). Then "22" from A32, then "9 23" from A33 if A33->A14 23 with A14->9. So that fits: A34->1 (simple), A32->22 (then A33-> A14 23 with A14->9). So A31 could generate "1 22 9 23". Good. Then after that we have tokens "21 20" (which likely come from A38 or from following A38). Actually rule_49: A29 -> 1 A31 A38. So after A31 we have A38.

Thus A38 must generate "21 20 ..." plus further tokens: after "21 20" we have token 30 = 11, token31=1, token32=14, token33=10, token34=1, token35=1, token36=5, token37=8, token38=3, token39=1, token40=4, token41=2.

Thus A38 must produce "21 20 11 1 14 10 1 1 5 8 3 1 4 2" (since after A31). Let's check A38 production: rule_64: A38 -> A39 20.

Thus A38 yields [A39] then terminal 20. So final token of A38 is 20, matching target's token 29 = 20 (the second token of that segment). In target, we have "21 20". So that would be A39 yields 21, then we have the 20 token from rule_64. This matches. So A38 is appropriate.

Now after token 20, the next token in target is 11 (position 30). But there are no further symbols in A38 after terminal 20. So after A38, we go back to A16? Wait, A29 is a child of A1; after A29 we have A16 and then A2. So after A29 (including its A38) we will have A16 (next). So after the 20, we will move to A16. That's consistent: after 20, the target's token 30 is 11, which would be generated from A16.

Thus we need to parse A16 to produce "11 1 14 10 1 1 5 8 3 1 4 2". Let's inspect A16's productions.

A16 rules:

rule_23: A16 -> 1
rule_24: A16 -> 1 A17 A22 10
rule_25: A16 -> A18 A17
rule_26: A16 -> A19 A17 10
rule_27: A16 -> A20 A17 10
rule_28: A16 -> A26 A17 14 10

Thus must produce a sequence starting with 11 maybe. But the first token target for A16 is 11. None of the RHS start with 11 directly except maybe via A18 (if A18 -> 11). Indeed rule_32: A18 -> 11. So maybe we use rule_25: A16 -> A18 A17. Then A18 yields 11. Then A17 yields something.

Thus we can produce "11" at start via A18. Then A17 may produce the rest.

Now A17 rules:

rule_29: A17 -> 1
rule_30: A17 -> 1 A21

Thus A17 can produce "1" or "1 A21". A21 rules: rule_35: A21 -> A18; rule_36: A21 -> A19; rule_37: A21 -> A20. So A21 yields either A18 (which could be 11 or 1) or A19 (producing 1) or A20 (producing 1). So A21 can produce a terminal sequence "1", "11", or maybe other? Let's explore.

Thus A16 via rule_25 yields sequence: [A18] [A17] => possibly "11 1" if A17 -> 1; or "11 1 A21" if using rule_30.

Our target after 11 is "1". So initial part "11 1" works. Following that, we need "14 10 1 1 5 8 3 1 4 2". The rest may come from A21 expansions and subsequent insertions.

But A21 expansions: A21 could be A18 (which could produce 1 or 11), but we need 14 next. There's a terminal 14 in the target. Where does 14 appear in productions? Several places: rule_21: A14 -> 9; rule_21 actually A14->9. Let's find rule for terminal 14: rule_28: A16 -> A26 A17 14 10 includes terminal 14 after A17. Also rule_28 includes terminal 14. Terminal 14 appears as a literal there. Also rule_28 includes terminal 10. So perhaps those are used later.

We have target segment "14 10". That's tokens 32:14, token33:10. So maybe we can produce these via rule_28 (A16 -> A26 A17 14 10). However our current A16 path is using rule_25 (A18 A17). That doesn't produce 14 and 10. So maybe we need a more complex A16 derivation to produce the remainder "14 10 ..." after initial "11 1".

But maybe we can do mix: A16 -> A26 A17 14 10 - the RHS is: A26, A17, 14, 10. This would produce a sequence that includes a 14 and 10 at the end. The beginning would be whatever A26 yields. Then A17 yields something, then 14 and 10. So maybe the first token of A16 could be "11"? Let's see if A26 can generate 11. A26 rule: A26 -> A27 11 (rule_43). A27's production: rule_44: A27 -> A28 1 16 (A28 then 1 then 16). So A27 yields A28 1 16. A28 can be 1, 18, 19. So A26 yields: (A28 1 16) 11. That's "A28 1 16 11". This would start with maybe 1 (if A28->1) or 18 or 19. Not 11. So not 11.

Alternatively, A26 -> A27 11 yields could produce "1 1 16 11". Not 11 at start.

Thus not ideal.

Alternatively, A16 rule_24: A16 -> 1 A17 A22 10. That includes a leading 1. Not 11.

Alternatively, A16 rule_26: A16 -> A19 A17 10. A19 can be 1. Not 11.

A16 rule_27: A16 -> A20 A17 10. A20 is 1.

Thus the only rule that includes terminal 11 at the start is rule_25 (A18 A17) where A18 can be 11. Good.

Thus we may need to continue after A17 to generate the rest tokens. However A16 from rule_25 ends after A17, no more terminals. So to generate 14 and 10 we must incorporate them via insertions into either A17's productions or after A16? Insert terminals anywhere in RHS of any used rule. Since rule_25's RHS is "A18 A17". We can insert terminals after A17 (i.e., before finishing the rule) because we can insert terminals anywhere in the RHS. For instance, modify rule_25 to "A18 A17 14 10". That would be equivalent to rule_28 but with A26 replaced by A18? Actually rule_25 is "A16 -> A18 A17". Inserting "14 10" after A17 yields A16 to produce "A18 A17 14 10". That would generate "11" (if A18 ->11), then A17 yields "1", then 14, then 10. That matches "11 1 14 10". Good. This could replace the need for using rule_28.

Thus we can insert two terminals (14 and 10) into rule_25. However 10 already appears later in target after 14; that's fine.

Thus modifications: rule_25 (A16 -> A18 A17) insert terminals 14 and 10 after A17. Or we could use rule_28 which already includes 14 and 10 but involves A26 and A17 before. That may be more complex.

Thus define: modify rule_25: A16 -> A18 A17 14 10. Insert "14 10". That's 2 insertions.

Now we need to produce rest of tokens after 10: after we've produced 11 (A18->11), A17 -> maybe produce "1 A21"? likely we need to generate more tokens after that: after the "10", we still need "1 1 5 8 3 1 4 2". Let's check target after tokens we've accounted for:

We used: tokens 30=11 (A18->11), token31=1 (A17 -> 1?), token32=14 (insert), token33=10 (insert). The next tokens token34=1, token35=1, token36=5, token37=8, token38=3, token39=1, token40=4, token41=2.

Thus after 10 we need "1 1 5 8 3 1 4 2".

We haven't accounted for these after A16; after A16 we go to A2 (last nonterminal). So these leftover tokens must be derived from A2.

Thus A2 must generate "1 1 5 8 3 1 4 2". Let's examine A2 productions again:

rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A6
rule_4: A2 -> A4 A3 A5
rule_5: A2 -> A12 A3 A11

Thus many possibilities.

We need to produce sequence of length 8: 1,1,5,8,3,1,4,2.

We see maybe we can use rule_4: A2 -> A4 A3 A5. Terminals: from A4 (1 or 3), from A3 (1), from A5 (4 2). That yields something like: [A4] 1 (A3) 4 2.

If we select A4->1 then we get "1 1 4 2". That's tokens: 1, 1, 4, 2. But we need additional tokens "5 8 3" placed between maybe? Actually we need "1 1 5 8 3 1 4 2". That includes "5 8 3 1". So might need expansions of A5 to produce "5 8 3 1 4 2"? But A5 is only "4 2". So can't produce extra. Could we insert terminals into A5's rule (rule_9: A5 -> 4 2) to produce "4 2"? We could insert "5 8 3 1" before "4 2"? That would be 4 extra terminals inserted. That would produce "5 8 3 1 4 2". That would give A5 outputs "5 8 3 1 4 2". Then A2 via rule_4 with A4->1 and A3->1 would yield "1 1 5 8 3 1 4 2". That matches exactly. So modifications: Insert terminals "5 8 3 1" before the "4 2" in rule_9. Actually the rule is "A5 -> 4 2". Insert "5 8 3 1" before maybe after 4? The order. Our needed sequence for A5 is "5 8 3 1 4 2"? Wait we need "5 8 3 1 4 2". However note that after "1" we need "4 2". So we want A5 -> 5 8 3 1 4 2. That is we need to add "5 8 3 1" before the existing "4 2". Insert four terminals.

Thus we can modify rule_9 to be "A5 -> 5 8 3 1 4 2". That is 4 insertions.

Now choose A4 -> 1 (via rule_7). A3 -> 1 (via rule_6). So A2 becomes: A4 (1) A3 (1) A5 (5 8 3 1 4 2) => sequence "1 1 5 8 3 1 4 2". Exactly as target.

Thus we need to insert 4 terminals in rule_9.

Now overall insert count: earlier we had insertions:

- rule_68: 3 terminals (16 1 17)
- rule_97: 2 terminals (27 31)
- rule_106: 1 terminal (25)
- rule_86: 1 terminal (13)
- rule_25: 2 terminals (14 10)
- rule_9: 4 terminals (5 8 3 1)

Total so far: 3+2+1+1+2+4 = 13 insertions.

Potentially also need to address any discrepancy for A29 generation of tokens after 28 and 6 etc.

We haven't examined A29's generation thoroughly. Let's parse that part.

We have chosen to use rule_50 (A29 -> A9 A31 A30) with an insertion of 28 before A9 (to generate initial 28 token). We also need A9->6 (via rule_14). So rule_50 with insertion becomes: A29 -> 28 A9 A31 A30? Wait insertion could be before A9. Actually rule_50's RHS is: A9 A31 A30. So we can insert "28" before A9, "28 A9 A31 A30". That yields tokens: 28 [expansion of A9] ... Good.

Alternatively, we could have inserted on A9's rule to include 28, but easier to insert on rule_50.

Thus we need to insert terminal 28 before A9 in rule_50. That's 1 insertion.

Now we must see if the rest matches target.

Now after A9 yields 6, then A31 A30 must generate the tokens "1 22 9 23 21 20". Let's test.

We have target after token 23=6 as tokens: token24=1, token25=22, token26=9, token27=23, token28=21, token29=20.

Thus A31 should generate "1 22 9 23". A30 generates "21 20". Indeed A30 rule: A30 -> 21 20 (rule_52). Perfect.

Thus for A31 we need to derive "1 22 9 23". Let's examine A31.

Options:

- A31 -> 1 (rule_53) then A30? Wait rule_49 inserted earlier? Actually A31 expansion is just the first part (A31 -> 1) yields "1". That would not produce rest of needed tokens.

- A31 -> A34 A32 (rule_54). This yields A34 then A32.

Thus we need A34 A32 => "1 22 9 23". Let's see.

We need to produce "1" from A34, and "22 9 23" from A32.

A34 can be either "1" (rule_58) or "A37 A35" (rule_59). A37 always yields "1". So whichever yields at least a "1". So we can use A34 -> 1 (simple). Then A32 must generate "22 9 23". A32 rule: A32 -> 22 A33 (rule_55). So we need A33 to generate "9 23". A33 options: A33 -> A10 (rule_56) yields "1" or "7". Not suitable. Or A33 -> A14 23 (rule_57). A14 can be 1 (rule_20) or 9 (rule_21). So if we choose A14->9, then A33 -> 9 23. Perfect. So A32 => 22 + (9 23) = "22 9 23". Good.

Thus A31 -> A34 A32 with A34->1 (rule_58) and A32 as above yields "1 22 9 23". So A31 matches.

Thus nothing missing for A31/A30. No insertions needed.

Now after A31 we have A30 which yields "21 20". That's good.

Thus A29 (rule_50) with insertion yields: "28" plus expansion of A9 (6) plus A31 (1 22 9 23) plus A30 (21 20) => "28 6 1 22 9 23 21 20". That matches target tokens 22-29: 28 6 1 22 9 23 21 20. Great.

Thus we need 1 insertion in rule_50.

Now after that, we go to A16 (already handled) and then A2.

Thus we've covered the whole target.

Now just verify that all needed tokens have been accounted for:

Sequence breakdown with modifications:

A1: A53 A45 A29 A16 A2

A53 (rule_95 with modifications):
- rule_95: A53 -> A60 A54 30 (no changes needed besides A60, A54 modifications)
- A60 (modified rule_106: A60 -> A61 25 19)
-- A61 (rule_107): A10 A37 A62 -> 7 1 24 (choose A10->7)
-- So A60 yields "7 1 24 25 19"
- A54 (modified rule_97 and rule_68):
-- A54 -> A40 26 A55 27 31 (insertion of "27 31")
-- A40 (modified rule_68: "1 16 1 17")
-- So A40 yields "1 16 1 17"
-- Terminal 26
-- A55 (choose A23 -> 15) yields "15"
-- Inserted "27 31"
Sequence: "1 16 1 17 26 15 27 31"
Thus A53 yields the concatenation: A60 part "7 1 24 25 19" + A54 part "1 16 1 17 26 15 27 31" + terminal 30 => So full A53: "7 1 24 25 19 1 16 1 17 26 15 27 31 30". Indeed matches tokens 1-14.

Now A45 (using rule_79 with insertion in rule_86):
-- A45 -> A49 A46 29 (rule_79)
-- A49 (modified rule_86: A49 -> A50 13 18)
--- A50 (rule_87): A18 1 12, choose A18->11 -> yields "11 1 12"
--- Insert 13 before terminal 18: So A49 yields "11 1 12 13 18"
-- A46 (choose rule_81: A46 -> 1)
-- terminal 29

Thus A45 yields "11 1 12 13 18 1 29". That matches tokens 15-21.

Now A29 (rule_50 modified):
-- Insert 28 before A9: A29 -> 28 A9 A31 A30
-- A9 -> 6 (rule_14)
-- A31 (via rule_54: A34 A32)
--- A34 -> 1 (rule_58)
--- A32 = 22 A33; A33 -> A14 23 (rule_57) with A14 -> 9 (rule_21)
Thus A31 yields "1 22 9 23"
-- A30 -> 21 20
Thus A29 yields "28 6 1 22 9 23 21 20". Tokens 22-29.

Now A16 (modified rule_25):
-- A16 -> A18 A17 14 10 (inserted 14 10)
--- A18 -> 11 (rule_32)
--- A17 -> 1 (rule_29) [or could include A21 for extra, but we need just one 1]
Thus A16 yields "11 1 14 10". That's tokens 30-33.

Now A2 (using rule_4):
-- A2 -> A4 A3 A5 (rule_4)
--- A4 -> 1 (rule_7)
--- A3 -> 1 (rule_6)
--- A5 (modified rule_9): 5 8 3 1 4 2 (inserted 5,8,3,1)
Thus A2 yields "1 1 5 8 3 1 4 2". Tokens 34-41.

Thus entire target sequence is derived.

Now let's count insertions we made:

List modifications:

- rule_68: inserted terminals 16, 1, 17 (three separate terminals). Actually inserted 16, 1, 17: note there is already a 1 before; the original rule is "A40 -> 1". So after 1 we need "16 1 17". That's three insertions.

- rule_97: inserted terminals 27 and 31 after A55. So "A40 26 A55 27 31". Two insertions.

- rule_106: inserted terminal 25 between A61 and 19. One insertion.

- rule_86: inserted terminal 13 before 18. One insertion.

- rule_25: inserted terminals 14 and 10 after A17. Two insertions.

- rule_9: inserted terminals 5,8,3,1 before the existing 4 2. Four insertions.

- rule_50: inserted terminal 28 before A9. One insertion.

Total = 3+2+1+1+2+4+1 = 14? Wait earlier we counted 13. Let's recount: 3+2=5, +1=6, +1=7, +2=9, +4=13, +1=14. Yes 14 insertions.

Check: Did we include any other modifications? We modified rule_95? No, unchanged. rule_95 remained same. rule_68, rule_97, rule_106, rule_86, rule_25, rule_9, rule_50.

Thus total 7 rules changed.

Now check if any other insertions needed for A31's derivations? No. A32 uses rule_55: 22 A33. No missing. A33 -> A14 23: we need A14 to be 9, which is allowed by rule_21; no insertion needed.

Potentially verify that A30 is just "21 20", matches tokens 28 and 29: yes.

Now check any potential missing token for A16 after A17: we inserted 14 and 10; but our A16 gets "A18 A17 14 10". That yields after 11 and 1: we get 14 and 10. Confirm target after 11, 1 we have 14 then 10. Yes.

Now check A17 used 1 via rule_29, not A21 as we didn't need extra. Good.

Now check A18 via rule_32 is 11 which matches token. Yes.

Now A45 after that we used A49 with inserted 13: Let's verify A49 rule: original rule_86: A49 -> A50 18. So we inserted 13 before 18 making "A50 13 18". That's fine.

Now A50 -> A18 1 12, where A18->11 yields "11 1 12". Inserted 13 makes "11 1 12 13". Then terminal 18 yields "11 1 12 13 18". Good.

Now A46 via rule_81 yields "1". Good.

Now A45 final token is 29 (from rule_79). So A45 yields "11 1 12 13 18 1 29". Good.

Now after A45 we have token 28 from A29 insertion.

Now A29 rule_50 insertion yields "28 A9 A31 A30". Good.

Check A9 is 6 (via rule_14). Good.

Thus we accounted for all tokens.

Let's double check sequences for any extra or missing.

Full sequence assembled:

A53: [7 1 24 25 19] [1 16 1 17 26 15 27 31] 30
Thus tokens: 7,1,24,25,19,1,16,1,17,26,15,27,31,30.

A45: 11,1,12,13,18,1,29

Thus tokens: 11,1,12,13,18,1,29.

A29: 28,6,1,22,9,23,21,20

A16: 11,1,14,10

A2: 1,1,5,8,3,1,4,2

Combine:

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

Matches exactly the target.

Thus our modifications succeed.

Now we need to output only the updated production rules with rule identifiers.

We must list each rule that changed: rule_68, rule_97, rule_106, rule_86, rule_25, rule_9, rule_50.

But also rule_95? No changes. We haven't changed rule_95. rule_25 is "A16 -> A18 A17". We modify to "A16 -> A18 A17 14 10". We'll write that.

Now check that rule_25 originally: A16 -> A18 A17. We'll rewrite as "A16 -> A18 A17 14 10". Good.

Now rule_68: A40 -> 1 (original). Modified: "A40 -> 1 16 1 17". (One might need to ensure spacing: Insert terminals after 1.)

Now rule_97: original "A54 -> A40 26 A55". We modify to "A54 -> A40 26 A55 27 31". Insert 27 and 31 after A55.

Now rule_106: originally "A60 -> A61 19". Modified to "A60 -> A61 25 19". Insert 25 before 19.

Now rule_86: originally "A49 -> A50 18". Modified to "A49 -> A50 13 18". Insert terminal 13 before 18.

Now rule_9: original "A5 -> 4 2". Modified to "A5 -> 5 8 3 1 4 2". Insert 5,8,3,1 before the existing 4 2, or maybe we insert after 4 2? Wait original has "4 2". To get "5 8 3 1 4 2", we need to insert "5 8 3 1" before the "4 2". Alternatively, we could insert after 4 2 but that would produce "4 2 5 8 3 1". That would not match target. We need them before. So modify to "A5 -> 5 8 3 1 4 2". perfect.

Now rule_50: original "A29 -> A9 A31 A30". Modified to "A29 -> 28 A9 A31 A30". Insert 28 before A9.

We also modified rule_68 and rule_97.

Double-check for consistency: rule_68 is "A40 -> 1". Changing to "A40 -> 1 16 1 17". That's okay.

Now check that A40 is used in rule_97; after modification, it yields "1 16 1 17". Good.

Now check if any other rule uses A40 (like rule_97 only). That's fine.

Now rule_97 is used in A54, which in turn is used in A53. Good.

The inserted tokens (27 31) after A55 will produce those tokens before the final 30.

Works.

Now rule_106 inserted 25 before 19.

Now rule_86 insertion 13 ensures A49 yields "A50 13 18". Let's recalc: A50 -> (A18 1 12). With A18->11 yields "11 1 12". Append 13 then 18 yields "11 1 12 13 18". That's correct.

Now rule_25 insertion of 14 10 after A17 yields A16 -> A18 A17 14 10. A18->11 yields 11; A17->1 yields 1; then 14 10 yields correct.

Now rule_9 insertion 5 8 3 1 before 4 2 yields A5 -> 5 8 3 1 4 2.

Now rule_50 insertion of 28 yields correct.

Thus all modifications are correct.

Now we need to output "Updated production rules:" then each rule line with modifications.

We must include the rule number (e.g., rule_68:) with the new RHS.

Be careful formatting: Should be exactly like in original list, with colon after rule number and spaces. Eg:

rule_68: A40 -> 1 16 1 17

No preceding or trailing spaces.

Thus final answer should include only updated rules.

List modifications:

- rule_68
- rule_97
- rule_106
- rule_86
- rule_25
- rule_9
- rule_50

Make sure to maintain same rule naming but with updated RHS.

One may also consider rule_68's order of terminals: could be "1 16 1 17". Good.

Now also we inserted 27 and 31 in rule_97. Should be placed after A55. So "A54 -> A40 26 A55 27 31". Might want to follow same style: rule_97: A54 -> A40 26 A55 27 31

Now rule_106: "A60 -> A61 25 19"

Now rule_86: "A49 -> A50 13 18"

Now rule_25: "A16 -> A18 A17 14 10"

Now rule_9: "A5 -> 5 8 3 1 4 2"

Now rule_50: "A29 -> 28 A9 A31 A30"

Now verify that there are no other dependent insertions needed.

Potentially need to ensure rule_68 modifications don't violate that A40 may also be used elsewhere (like rule_97) only; fine.

Now output accordingly.

Thus the final output is:

Updated production rules:
rule_68: A40 -> 1 16 1 17
rule_97: A54 -> A40 26 A55 27 31
rule_106: A60 -> A61 25 19
rule_86: A49 -> A50 13 18
rule_25: A16 -> A18 A17 14 10
rule_9: A5 -> 5 8 3 1 4 2
rule_50: A29 -> 28 A9 A31 A30

Make sure to include newline after each.

Potentially we also changed rule_68? Yes.

Double-check rule numbers: "rule_68" corresponds to "A40 -> 1". Yes.

"rule_97": yes.

"rule_106": yes.

"rule_86": yes.

"rule_25": yes.

"rule_9": yes.

"rule_50": yes.

Thus final answer.

Before concluding, check if any other possible missing terminal symbols appear in A45 expansions beyond those we accounted. For instance, A45 may produce A46 (which we used rule_81) which yields 1. That's fine. No missing tokens.

Check A45 final token is 29; good.

Now A45's first part A49 yields 11, etc. Did we need to produce token 13? Inserted.

Now check A45 also uses token 29 which appears as terminal 29; good.

Now after A45, we have A29 produce token 28 inserted.

Now check that target includes token "28" after 29. Yes.

Thus overall.

Now verify A45's rule_79 unchanged except for insertion in rule_86.

Now check A5 rule: we inserted 5 8 3 1 before 4 2. This yields output "5 8 3 1 4 2". Our target's suffix after A2 begins with "1 1 5 8 3 1 4 2". We have A4 produce 1, A3 produce 1, then A5 produce "5 8 3 1 4 2". That's correct.

Thus final count of insertions: 3 + 2 + 1 + 1 + 2 + 4 + 1 = 14. Verify no other insertions needed.

Now let's double check there is no conflict such as using rule_68 with inserted 1 16 1 17 may cause duplicate terminals 1? That's fine.

Check that rule_68 is used only within rule_97. That's fine.

Thus final answer is good.

Will output lines exactly as required.

Make sure not to write anything after updated production rules. The instruction: "Do not write anything after the updated production rules."

Thus output only those lines.

Will prepare final answer accordingly.Updated production rules:
rule_68: A40 -> 1 16 1 17
rule_97: A54 -> A40 26 A55 27 31
rule_106: A60 -> A61 25 19
rule_86: A49 -> A50 13 18
rule_25: A16 -> A18 A17 14 10
rule_9: A5 -> 5 8 3 1 4 2
rule_50: A29 -> 28 A9 A31 A30