We need to parse the problem: We have a CFG with certain production rules, but some terminal symbols have been removed. We need to insert terminals back into production rules (anywhere) to enable generating the target sequence from start symbol A1.

We must minimize total number of terminal insertions. Deleting or reordering symbols, or creating new production rules, is prohibited.

We need to propose where to insert terminals (maybe multiple places) into existing production rules, not adding new rules. So we can only augment existing LHS→RHS sequences with extra terminals.

Goal: ensure there exists a parse generating the exact target sequence. Need to find a derivation path from A1 → ... → string = target.

We have many rules, likely already produce something similar, maybe missing some terminals.

We need to consider candidate derivations that could produce the target sequence (which is 34 numbers?). Let's count them: The target sequence is:

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

That is 34 symbols.

We have terminals 1-32. The target uses each at least once. Actually I see 31,27,23,13,15,22,16,12,17,19,20,14,13 (repeat),1,18,26,10,8,11,30,32,9,29,24,7,21,4,6,5,2,3,25,28,27 (27 appears again at end). Okay.

The given CFG includes many rules that generate sequences of terminals and nonterminals. Probably the target can be generated by one of the A2 rules (since A1 -> A2). A2 rules produce a long sequence containing many terminal symbols: e.g., rule_2: A2 -> 31 27 A54 A61 A8 30 A8 A60 27

Let's see: This yields 31 27 ... 30 ... 27, which aligns with start and end of target: target begins with 31 27 and ends with ... 27 but not exactly just 27 at end? Actually target ends with 27, and starts with 31 27, so these appear.

So rule_2 seems promising: 31 27 A54 A61 A8 30 A8 A60 27

Let's inspect the substructures: A54 appears; A61 appears; A8 appears a few times; A60 appears. So we need the expansions of A54, A61, A8, A60 to generate the rest of sequence: after 31 27 we need: 23 13 15 22 16 12 17 19 20 14 13 1 18 26 10 8 11 30 ... Wait after the second token 27, the next tokens are 23 13 15 22 16 12 17 19 20 14 13 1 18 26 10 8 11 30... Actually the target after 31 27 we have 23 13 15 22... So maybe A54 expands to 23 ... 18 etc. Let's examine.

A54 → A55 A14 (rule_97). A55 → 23 (rule_98). So A54 expands to 23 A14. So far we get 23 + whatever A14 produces.

A14 has two alternatives (rules 24-26): A14 -> 12 A15 or 13 A16. So A14 is either 12 A15 (which yields 12 + something) or 13 A16 (which yields 13 + something). Which one matches target after 23? Target after "23" is "13". So it's likely we need A14 -> 13 A16 (since target has 13 next). So we might want to choose rule_26: A14 -> 13 A16. Good.

Now A16 has productions: rule_29: A16 -> 1; rule_30: A16 -> 2; rule_31: A16 -> 15. So we could produce either 1, 2, or 15. Sequence after "13" is "15". So we need A16 -> 15 via rule_31.

Thus A54 expands to: 23 13 15. That matches target so far: 31 27 23 13 15.

Great.

Now after A54, rule_2 continues with A61. After our matched part, target sequence after "15" is "22". So A61 must expand to produce "22". Also later we need "16 12 17 19 20 14 13 1 18 26 10 8 11 30 ..." Actually we need to see the whole sequence: target:

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

Now, after A61, rule_2 says "A8". So after A61 we need to generate "16" then "A8" generates some more tokens after A61. Wait A61 may produce multiple tokens; then A8 produce something else before we get to "30". Actually in rule_2 we have: 31 27 A54 A61 A8 30 A8 A60 27. So after A61 we have terminal A8 (nonterminal), then 30, then another A8, then A60, then 27. So the sequence after A54 is: [expansion of A61] [expansion of A8] 30 [expansion of A8] [expansion of A60] 27.

Target after 31 27 23 13 15: we need 22 16 12 17 19 20 14 13 1 18 26 10 8 11 30 32 9 29 24 7 21 4 6 5 2 3 25 28 27.
We have a "30" at position 20 of final sequence (target). Indeed, after initial segment, eventually we have a 30, then 32 etc. So in rule_2, we have a single terminal "30" at that position. So we need to match that the 30 appears at the same position as expected. Let's see if that matches. Since after A61 and A8 there is a 30 (i.e., before 30, we have emission of A61 and then exactly one A8 expansion). So the 30 produced by rule_2 is at index approx after the expansions of A61 and first A8. We need to check alignment.

The target has 30 at position 20 (count from start). Let's count earlier:

Up to position 5 we matched: 31 (i1), 27 (i2), 23 (i3), 13 (i4), 15 (i5). Position 6 is 22, 7:16, 8:12,9:17,10:19,11:20,12:14,13:13,14:1,15:18,16:26,17:10,18:8,19:11,20:30,21:32,22:9,23:29,24:24,25:7,26:21,27:4,28:6,29:5,30:2,31:3,32:25,33:28,34:27.

Thus positions 6-19 need to be generated by A61 and the first A8 (which includes expansions). Then we have 30 at position 20, matches just after the first A8, then position 21 onward must be generated by second A8, then A60, then final 27.

Now second A8 appears after 30. In target, after 30 we have 32 9 29 24 7 21 4 6 5 2 3 25 28 27. So after 30 we have: 32,9,29,24,7,21,4,6,5,2,3,25,28,27 (14 tokens). The final 27 is after A60, not after second A8. Actually order: After 30, we have A8, then A60, then final 27. So A8 generates some prefix, then A60 generates some symbols, then final 27.

We need to see expansions of A8: there are two productions: (rule_16 A8 -> 9) and (rule_17 A8 -> A9 A10). So A8 can produce just terminal 9, or produce A9 A10 (nonterminals). A9 produces 10 (rule_18). A10 -> 8 A11 (rule_19). A11 can be 1,2,11. So A8 can produce two-element pattern: 9 (terminal) or 10 8 X where X is 1,2, or 11. Actually chain is A8 -> A9 A10, where A9=10, A10=8 A11. So yields 10 8 [A11]. So A8 either yields "9" (single token) or yields "10 8 [1|2|11]" (3 tokens). So A8 can produce at most 3 tokens (maybe 4 if A11 is 11? Actually A11 -> 11 is token 11 (single). So yields "10 8 11". So max 3 tokens.

Thus the first A8 will produce up to 3 tokens; second A8 also up to 3 tokens.

Now A61 definition: several alternative productions (rules_106-108). A61 -> A52 | A54 | A56. So A61 could expand to A52 or A54 (which we already used earlier) or A56.

A54 we already used earlier; producing 23 13 15. That's used earlier. If we use A61 -> A54 again, then we would produce again "23 13 15". However target next token after 15 is 22 (pos6). That doesn't match "23". So A61 likely using a different expansion such as A52 or A56, which produce longer sequences.

A52 defined: rule_101: A52 -> A53 A35 A12 A22. (only one production?). Indeed rule_95: A52 -> A53 A35 A12 A22 (makes sense). So A52 expands to A53 A35 A12 A22 (a concatenation).

A53 -> 22 A17 (rule_96). So A53 yields terminals [22] then A17.

A35 -> 19 A36 (rule_66). So A35 yields 19 then A36.

A36 -> 4 (rule_67) or 20 (rule_68). So A36 yields either 4 or 20.

A12 -> A13 A14 (rule_23). A13 -> 14 (rule_24). So A13 yields 14. Then A14 we already analyzed, could be 12 A15 or 13 A16. Later we need to produce the sequence that includes 12,17,19,20,14,13 etc.

A22 -> ... has many alternatives: A22 -> A23 | A24 | ... | A34 (rules 41-52). Each Ai (A23 … A34) all produce a 3 possibly (and some produce 18). Actually A23 to A32 produce 3; A32 also can produce 18; A33 and A34 produce 3. So A22 effectively expands to a single terminal (3) or 18 (if using A32->18). So A22 yields 3 (or 18) at its position.

Thus A52 expands to: (22 + expansion of A17) + (19 + (4|20)) + (14 + A14) + (3 (or 18)). So eventual sequence from A52: starts with 22, then something from A17, then 19, then 4 or 20, then 14, then something from A14, then 3 or 18.

Now A17 -> A18 A19 (rule_32). A18 -> 16 (rule_33). A19 -> 12 A20 (rule_34) OR 13 A21 (rule_35). So A19 yields either 12 A20 or 13 A21.

A20 -> 1 | 2 | 17 (rules 36-38). So yields 1,2,or 17.

A21 -> 1 | 2 (rules 39-40). So yields 1 or 2.

So A17 yields 16 + (12 or 13) + (some small). That could generate 16 12 1, 16 12 2, 16 12 17, 16 13 1, 16 13 2.

Good.

Now combine with A52: The sequence from A52: Term 22, then A17 yields 16 + (12 or 13) + [1/2/17], then 19, then either 4 or 20 (from A36), then 14, then A14 (again may be 12 A15 or 13 A16), then then terminal 3 or 18.

Thus the pattern: 22, 16, (12/13), (1/2/17), 19, (4/20), 14, (12/13 ...) (1/2/15?), then 3 (or 18).

Now compare with target after 31 27 23 13 15 (positions 1-5):
Positions 6 onwards: 22, 16, 12, 17, 19, 20, 14, 13, 1, 18, 26, 10, 8, 11, ...

Thus the next after 15 is 22 16 12 17 19 20 14 13 1 18. That matches pattern from A52 almost exactly: 22, 16, 12, 17, 19, 20, 14, 13, 1, 18... Let's see: from A52 we get 22, then A17 yields 16, 12, with A20 maybe produce 17 (since A20 -> 17 is allowed). Then after A17 we get 19, then A36 -> 20, then 14, then A14 (maybe choose production giving 13 A16 -> 13 then A16 maybe produce 1). Indeed A14 -> 13 A16; A16 -> 1, or 2, or 15. For required token sequence, after 14 we need 13, 1, 18. So A14 gives 13 1 if we pick A16 -> 1. Then after that, we need 18; A22 yields 3 normally, but A32 can produce 18 if we choose that branch: A22 -> A32 -> 18. Indeed A32 has productions 3 and 18. So we can get 18. Good!

Thus A52 can exactly generate the subsequence: 22, 16, 12, 17, 19, 20, 14, 13, 1, 18. Perfect.

Now after that, target continues: 26, 10, 8, 11, 30,...

Now we have after A61, we had A8 (first one) generate next tokens after A61, but we need to map where 26 occurs: In rule_2 after A61 we have A8 then 30. However we need 26 before 10. Actually after we have consumed A61 (which expands to A52) we have then A8 then 30. But before reaching 30 we need to produce tokens "26 10 8 11". The target: after 18 we have 26 10 8 11 then 30. So if A8 can produce "26"? No, A8 can produce only "9" or "10 8 X". Not 26. So the 26 token must be generated by something else, likely the first A8 (or part of A8) cannot produce 26. So perhaps the rule_2 chosen is not appropriate; maybe we should choose a different A2 rule which includes terminal 26 earlier. Let's see.

We have many A2 productions: rule_3,4,...9 etc. Let's list them:

- rule_2: A2 -> 31 27 A54 A61 A8 30 A8 A60 27
- rule_3: A2 -> 31 27 A54 A63 A8 A8 A62
- rule_4: A2 -> 31 27 A54 A65 A8 30 A8 A64
- rule_5: A2 -> 31 27 A54 A71 A8 32 A8 29 A70 28 27
- rule_6: A2 -> 31 A54 A59 A8 A8 29 A58 27
- rule_7: A2 -> 31 A54 A67 A8 A8 29 A66 28 27
- rule_8: A2 -> 31 A54 A69 A8 A8 A68 27
- rule_9: A2 -> 31 A54 A73 A8 A8 A72 27

These all begin with 31 at start; some have 27 after (rules 2-5). Rules 6-9 have 31 then A54 etc, no 27 after 31. The target begins with "31 27". So we need a rule that has 31 27 (immediately). That could be rule_2, rule_3, rule_4, rule_5 all have 31 27.

Nevertheless, after initial couple tokens we need "23 13 15" from A54 as we saw. So A54 is required early; all these have A54 following after 31 27. So that part matches.

Now after A54, we need a series that matches the rest of target. Rule_2 uses A61, then A8, then 30, then A8, then A60, then 27. Rule_3 uses A63, A8, A8, A62 (no 30 token). Let's examine rule_3: A2 -> 31 27 A54 A63 A8 A8 A62.

A63 is defined? rule_111: A63 -> A52 26. So A63 expands to A52 followed by 26. Good! That suggests that rule_3 can produce "A52 26" after the A54, maybe generating the 22.. etc then 26. And after A63 we have A8 A8 A62. So that could produce the remaining tokens: A8 ( first ), A8 ( second ), A62 ( final). Let's examine A62: rule_110: A62 -> A56. Actually A62 has two productions: rule_109? Wait we have rule_109: A62 -> A52 ; rule_110: A62 -> A56. Actually we have: rule_109: A62 -> A52; rule_110: A62 -> A56. So A62 could be A52 again or A56.

Now A56 is defined later: rule_99: A56 -> A57 A50 A3 A37. That's a huge sequence. That seems unlikely to produce the short remainder; maybe using A52 again is simpler. Let's examine A52 again: we saw it yields 22 ... 18 with 21 tokens? Actually length: approximate.

But note that after A63 and two A8's and finally A62 we still need to generate the tail of target after 26: target after 26 is 10 8 11 30 32 9 29 24 7 21 4 6 5 2 3 25 28 27. This includes 10,8,11,30 etc. The sequence after 26 begins with 10 8 11. That's exactly the expansion of A8 via the rule A8 -> A9 A10 which yields 10 8 [A11], and A11 could be 11. So we may produce 10 8 11 via one A8.

The target after 10 8 11 includes 30 32 9 29 24 7 21 4 6 5 2 3 25 28 27. This must be generated by remaining parts: perhaps A8 (the second A8) could produce 30 ??? Wait A8 cannot produce 30; but maybe A8 with other modifications? Actually A8 cannot produce 30 unless we insert terminal after A8 expansions (but we cannot modify rule definitions except to insert terminals). However we might insert a terminal "30" inside A8's production, but that's not allowed? The instruction says we can insert terminal symbols anywhere in existing production rules. So we could modify the rule A8 -> A9 A10 by inserting extra terminals anywhere, maybe after A10 (or before). However we must be careful to minimize insertions.

But perhaps an alternative is to use rule_3's A62, which maybe expands to A56, which could produce the rest of sequence after the two A8 expansions (including maybe 30 and following). But is A56 capable of generating our target? Let's analyze A56 through its definition:

rule_99: A56 -> A57 A50 A3 A37

Thus A56 expands to A57 A50 A3 A37.

A57 -> 24 A7 (rule_100). So A57 yields 24 then A7.

A7 -> 7 (rule_15).

Thus A57 yields 24 7.

A50 -> 21 A51 (rule_93). So A50 yields 21 then A51.

A51 -> 4 (rule_94). So A50 yields 21 4.

A3 -> A4 A5 (rule_10). A4->6 (rule_11). A5 -> 5 A6 (rule_12). And A6 -> 1 or 2 (rule_13,14). So A3 can expand to 6 5 X where X is 1 or 2. So A3 yields either "6 5 1" or "6 5 2".

A37 has many alternatives: A37 -> A38, A39, ..., A49 (rules 69-80). Each A38 through A49 produce a terminal 3 (or maybe other). Indeed each A38 -> 3, A39 -> 3,... A49 -> 3 (rules 81-92). So A37 expands to a single "3". So the overall A56 expands to: 24 7 21 4 (6 5 X) 3. Let's combine: 24 7 21 4 6 5 1 (or 2) 3. That's exactly a subsequence "24 7 21 4 6 5 1 3". But our target after 10 8 11 ... we have "30 32 9 29 24 7 21 4 6 5 2 3 25 28 27". So after 10 8 11, we have "30 32 9 29" before the pattern "24 7 21 4 6 5 2 3". So A56 (if used) could generate "24 7 21 4 6 5 2 3". Indeed A6 can yield 2, giving the needed "2". So that matches the subsequence "24 7 21 4 6 5 2 3". Good. So A56 provides the part after "... 29". However we also need "30 32 9 29" before that. This could be produced by A8 (second A8) maybe produce 30 32 9 29? But A8 can't produce terminals 30,32,9 directly except with insertion. But maybe we can use A8 -> 9 (terminal) for the "9". We also need a 30 and a 32. Perhaps we can insert these terminals into expansions of A8 or other nonterminals.

Let’s inspect possible placements where we might need to insert missing terminals.

Recall rule_3: A2 -> 31 27 A54 A63 A8 A8 A62

A63 -> A52 26 (rule_111). So after A54 we get A63 which yields A52 then 26. That matches 22... 18 26 sequence.

Now after A63 we have A8 A8 then A62.

Now, A62 may be A52 or A56 (or A52). For the tail after both A8's we need "30 32 9 29 24 7 21 4 6 5 2 3 25 28 27". Let's see if A62 can generate the suffix "24 7 21 4 6 5 2 3 25 28 27"? A56 yields "24 7 21 4 6 5 2 3". Then we still need "25 28 27". The remainder after A56 may be produced by something else like maybe using A62 as A56 continues after A8's. The leftover tokens "25 28 27" could be produced by something else, perhaps from some other nonterminal that appears after A62? However after A62 there is no additional nonterminals in rule_3; the rule ends. So whatever A62 expands to must generate the rest of the string up to the final "27". The final 27 is part of the target and appears at the end. In rule_3 there is no terminal 27 after A62; missing. That suggests we need to insert a terminal 27 after A62 in that production rule to match final 27. However we cannot create new productions but we can insert terminals. Indeed rule_3 currently ends with A62. In the target, after everything else there is a final 27. So we can insert terminal "27" after A62 in that rule (i.e., update rule_3 to "A2 -> 31 27 A54 A63 A8 A8 A62 27") with one insertion.

But need to also generate the 25 and 28 before final 27; can they be produced elsewhere? Let's examine possible expansions of A8 and A62 that could produce those.

We must generate sequence "30 32 9 29" and also later "25 28" before final 27.

Consider possibility that we use A8 to generate 30 (and maybe 32). However A8's production cannot generate 30 or 32 presently. Could we insert terminal for each of those into A8's productions? For minimal insertion, we can maybe insert terminal 30 before the RHS of A8 (or after) to produce 30 in the string. Let's see A8 productions:

- rule_16: A8 -> 9
- rule_17: A8 -> A9 A10

We could insert terminal '30' before the RHS in rule_16 or after? For minimal, we need to produce "30" at a certain point in the string: after the first set (the A52 part) we have A8 (first) and then A8 (second) before A62. The target after ... 26 is "10 8 11 30 32 9 29". Wait careful: after 26 we have "10 8 11 30 32 9 29". Actually we earlier said target after 26 is "10 8 11 30 32 9 29". Let's double-check. Full sequence after 26: we had "10 8 11 30 32 9 29 24 7 ..." Yes. Actually target list: ... 22 16 12 17 19 20 14 13 1 18 26 10 8 11 30 32 9 29 24 7 21 4 6 5 2 3 25 28 27.

Thus after 26, we start with 10,8,11 must come from A8 maybe (first A8). So first A8 we likely want to produce "10 8 11". Good.

Next token after that is 30, which does not fit in first A8. This could be second A8 producing 30 (maybe via insertion). Then we need "32 9 29" before A62 (which produces 24...). So perhaps second A8 could generate "30 32 9 29". That's four terminals, which is more than A8 can produce normally. Could we insert three terminals into that rule, across the production? For minimal, we may need to insert "30 32" after the A8 nonterminal expansion or before? Let's think.

Option: Insert "30 32" at end of second A8's production (either rule_16 or rule_17). Example: modify rule_17: A8 -> A9 A10 30 32 (that's inserting two terminals after A10). That would cause A8 to produce A9 A10 then 30 32, yields "10 8 X 30 32". X is from A11. If A11 => 11, we get 10 8 11 30 32. Good! Then need "9 29" after that before moving on to A62. But currently after second A8 we have A62 directly. So we could insert "9 29" before A62 (or after A62?) Actually the sequence after the second A8 should be "9 29" then A62 should produce the rest. So we can insert "9 29" before A62 (just after we added 30 32 inside A8). Alternatively we could insert "9 29" after A62 but that would be after the rest, not correct. So we need to insert "9 29" after the second A8 and before A62. But we can do that by inserting terminals into the RHS of rule_3 after second A8 and before A62. If we edit rule_3: A2 -> 31 27 A54 A63 A8 A8 9 29 A62 27 (in addition to insert 30 32 earlier). However note that "9 29" we might also generate via other rules? But direct insertion is presumably minimal.

But consider that A8 already includes the ability to produce 9 (via rule_16), but we need a "9" after 30 32. Perhaps we can use the first A8 for 10 8 11, and second A8 produce 9 via rule_16 (producing 9) and we can add "30 32" before that 9? Let's try to minimize insertions.

Goal string segment: after 26: "10 8 11 30 32 9 29". If we use first A8 to produce "10 8 11", we can let second A8 produce "9" via rule_16 (produces "9"). Then we need to produce "30 32" before that 9. However we can also insert "30 32" before using the second A8, i.e., right before the second A8 in rule_3's RHS. So we could modify rule_3: after second A8, maybe we want terminals "30 32" before the "9". But we have second A8 then A62. Actually we need "9". The "9" could be either the second A8 (if using rule_16) or could be inserted somewhere else; but there is also "9" after 32, then "29". So we could do: after second A8, insert "30 32", then second A8 (producing "9") then insert "29" before A62. That's one insertion of "30 32" as a sequence of two terminals; we also need to insert a "29". That's total three terminal insertions (30,32,29), unless we want to split them across multiple rules.

Alternatively, we could modify the second A8's production to include "30 32" after its generated "9". That would be rule_16: A8 -> 9 30 32 maybe (insert two terminals after 9). Then we would have "9 30 32" produced by second A8, but the order in target is "30 32 9". So that's not correct.

Alternatively, modify rule_17 (A8 -> A9 A10) to also include "30 32" after A10 and then maybe also add "9 29"? Actually order: we need "10 8 11" then "30 32 9 29". If we modify A8 (the second occurrence) to produce "10 8 11 30 32 9 29"? That would be a large insertion: we would need to insert "30 32 9 29" after the normal A9 A10 expansions (which produce 10 8 11). So inside rule_17 we could change: A8 -> A9 A10 30 32 9 29. That would generate "10 8 11 30 32 9 29". Perfect! That would remove need to insert terminals elsewhere; just insert four terminals into rule_17.

But note we also need to produce "30" token earlier? Actually our target after "10 8 11" is "30". So indeed "30 32 9 29" follow consecutively. So embedding all four terminals in the second A8's production makes sense.

Thus we could set:

- First A8 (the one preceding the second A8) produce 10 8 11 (via rule_17 using A9 A10 with A11=11). That's fine; no insertions required for first A8.

- Second A8 also use rule_17 (or rule_16). But we need to decide how to differentiate the two A8 instances: they both refer to same nonterminal A8, so they will both expand the same way unless we use nondeterministic choices: If A8 has multiple productions (rule_16 for "9" and rule_17 for "A9 A10"), we can choose a specific one for each occurrence. Since the grammar allows nondeterministic choice: each occurrence of A8 can be replaced by any of its productions. So we could have first occurrence use rule_17 (default) to produce 10 8 X (X=11). Second occurrence could also use rule_17, and we can have inserted terminals after A10 to produce additional needed terminals. Since rule_17 is "A8 -> A9 A10". We can insert terminals after A10 in the RHS (i.e., modify rule_17 globally). That would affect both occurrences, not just the second. That would cause the first A8 to also generate extra terminals (30 32 9 29). That's not desired, but maybe we can use rule_16 for the first A8, and rule_17 for the second A8? However first A8 we need to produce 10 8 11; rule_16 generates only "9". So we need first A8 to be rule_17; which yields 10 8 X. So the first A8 must be rule_17. If we modify rule_17 to add extra terminals after A10, then first A8 would also produce those extra terminals, which would cause the string to have "10 8 11 30 32 9 29" earlier, which doesn't match our target (target after first A8's expansion should only be "10 8 11").

Thus side-effect issue: Since we cannot create new rules, but we can add terminals inside rule_17 globally. That will affect all expansions of A8 across the grammar, not just one occurrence. However we can still choose for each occurrence which rule to apply: we can use rule_16 for one occurrence (producing just 9) and rule_17 for the other (modified). Let's think: Our desired pattern is "10 8 11" first; for that, we must use rule_17 with A11 -> 11. If we modify rule_17 to have extra terminals after A10, then even that first A8 will produce extra parts after "10 8 11". That would be problematic.

Alternative: Use rule_16 for the first A8 (to produce "9") but that doesn't match. So we cannot use rule_16 for first occurrence.

If we cannot globally modify rule_17 without affecting the first A8, we could instead modify rule_16 (the "9" production) to produce "10 8 11"? No, we need first A8 to produce the pattern; we could also modify rule_16 and rule_17 integrally: make rule_16 produce "10 8 11". But rule_16 is "A8 -> 9". If we insert terminals into it, we could replace "9" with "10 8 11"? However we cannot delete "9". The insert operation can add terminals anywhere; we can't delete "9". So we'd produce "9 10 8 11". That's wrong, order and extra 9.

Alternatively, we could leave rule_17 as is but modify rule_10 or other rules to insert after A8's evaluation. But think: after first A8, we need "30 32" later but before second A8. Actually the target order: "10 8 11 30 32 9 29". So perhaps we could use the second A8's production to give only "9". Then we need to insert "30 32" between the two A8's. That could be done by inserting "30 32" directly in rule_3 between the two A8 symbols. So rule_3 currently: ... A8 A8 A62. After modifications, we could make it: ... A8 30 32 A8 A62. That would give the needed "30 32" after the first A8, before second A8 which yields "9". Then we also need a "29" before A62 (after second A8) to match target. That could be inserted after the second A8 (again in rule_3) before A62: ... A8 30 32 A8 29 A62. Then after A62 we need to produce "24 7 21 4 6 5 2 3 25 28 27". But we currently have A62 which can be A56 (which yields 24 7 21 4 6 5 2 3). That's good for the segment up to 3. However we still need "25 28". Those could be appended by insertion after A62 (i.e., after its expansion) before final 27 (inserted after A62 as earlier). Actually we earlier considered inserting final 27 after A62. We also need "25 28" before that. Observe that A56 doesn't produce 25 or 28. Perhaps we could add "25 28" as insertions after A62 (i.e., after either A56). So after A62 we would have inserted "25 28" before the final 27.

Thus, modifications needed:

- Insert "30 32" between the first A8 and second A8 in rule_3.
- Insert "29" after the second A8 (still before A62) in rule_3.
- Insert "25 28" after A62 (or after A62 but before final 27).
- Insert final "27" after A62 (or after inserted "25 28").

That's 2 + 1 + 2 + 1 = 6 insertions (two for "30" and "32" individually, one for "29", two for "25" and "28", one for final "27"). But perhaps we could combine "25 28 27" into a longer insertion but each terminal counts individually, so 3 insertions.

Total insertions: 2 + 1 + 3 = 6.

We need to check if we can reduce further by using existing terminal productions elsewhere to generate some of those missing tokens, instead of inserting them manually.

For "30": Is there any nonterminal that naturally yields "30"? Yes, rule_2 has a terminal 30. Also rule_4 has a 30. Not many others. A8 cannot produce 30. A60 maybe? Let's see the definitions: A60 -> A56 25 (rule_105). A56 is used elsewhere. A60 thus yields A56 then 25. That yields "24 7 21 4 6 5 X 3 25". Actually A56 yields "24 7 21 4 6 5 X 3". Then A60 yields that plus "25". So A60 yields the "24 7 21 4 6 5 X 3 25". That's close to the tail we need, but we also need "28 27". Also we need "30 32 9 29" before that. In original rule_2, after A8 we have terminal 30, then A8, then A60, then 27. So that gave "30" and "A8" and "A60". That seems to match the tail structure we need: A8 could produce "9" maybe, A60 yields the rest, and there's final 27. Indeed the original rule_2 design suggests "30" appears directly after first A8, then A8 produces maybe "10 8 11"? Actually in rule_2, after A8 we have a terminal 30, then A8 again, then A60, then 27. In this structure, we could produce: first A8 could be "10 8 11" (as needed), then 30 matches, then second A8 could be "9". Then A60 yields "24 7 21 4 6 5 2 3 25". That's matching "24 7 21 4 6 5 2 3 25". Then final 27 matches. The only missing pieces: "32" and "29" and "28". The target after "10 8 11" is "30 32 9 29 ..." Actually rule_2 includes a 30 but not 32 or 29 or 28. We could perhaps generate "32" and "29" and "28" via insertions elsewhere: maybe "32" could be from A8? No, 32 not produced. A30? Not present. 32 appears as terminal only in rule_5 (A2 -> ... A8 32 A8 29 A70 28 27). That rule uses 32 and 29 in some other positions. But our rule_2 doesn't have them.

Alternatively, maybe we could use rule_5 instead, which includes 32 and 29 and 28. Let's examine rule_5: A2 -> 31 27 A54 A71 A8 32 A8 29 A70 28 27.

This seems promising! This rule includes 32, 29, 28, and final 27. Let's check if the rest will match our target.

We need after A54 (which gave 23 13 15) to generate "22 16 12 17 19 20 14 13 1 18 26 10 8 11 30 32 9 29 24 7 21 4 6 5 2 3 25 28 27". Let's see.

In rule_5: after A54 we have A71, then A8, then terminal 32, then A8, then terminal 29, then A70, then terminal 28, then terminal 27. Also we lack terminal 30 before 32, and we lack 26. The target includes 26 before 10 8 11. Also includes 30 before 32. But rule_5 includes no "30" and no "26". So not perfect.

We could insert those missing tokens in rule_5: add "26" before the first A8 (or after A71?), add "30" before "32". Let's see the required ordering: after the A54, we need "22 16 12 17 19 20 14 13 1 18 26 10 8 11 30 32 9 29 24 7 21 4 6 5 2 3 25 28 27". In rule_5 we could produce "22 ... 18" via A71 (which probably expands similarly to A63), A71 defined? Let's examine A71: rule_124: A71 -> A52 26. Also rule_125: A71 -> A56. So A71 can expand to A52 26 (that gives the 22 ... 18 + 26). Good! So A71 will give the sequence of A52 then 26, which matches exactly needed "22 16 12 17 19 20 14 13 1 18 26". Good.

Then after A71, the next token needed is "10 8 11". That's the first A8 after A71. So we need first A8 to generate 10 8 11. That's done by using rule_17 for A8 with A11=11.

After that we need "30". In rule_5, after that A8 (which is after A71), we have terminal "32". Actually rule_5: ... A71 A8 32 A8 29 A70 28 27. So there is a terminal "32" before second A8. There's no "30". We need to produce "30" before "32".

We could insert "30" after the first A8 (i.e., after A8 and before terminal 32). So insert terminal 30 there.

After "32" we need "9" before "29". In rule_5, after terminal 32 we have A8. So that A8 can produce "9". Indeed we could use A8 -> 9 (rule_16) to generate the needed "9". Good.

After that we need "29". The rule already has terminal "29" after second A8. So good.

Now after "29" we need "24 7 21 4 6 5 2 3 25". In rule_5 after terminal 29 we have A70, then 28, then 27. Let's examine A70: rule_122: A70 -> A52 ; rule_123: A70 -> A56 25. So A70 can either be A52 (the sequence that ends with 18). Or A56 25 (which yields "24 7 21 4 6 5 X 3 25") but does that include the 25? Actually A56 yields "24 7 21 4 6 5 X 3". Then A70 -> A56 25 yields that combined with 25, giving "24 7 21 4 6 5 X 3 25". That's exactly needed after 29: we need "24 7 21 4 6 5 2 3 25". Where X should be 2. In A56, there is A6 which yields 1 or 2 (or also 2 for the 2 we need). So we can pick A6 -> 2. Good. So A70 can be A56 25 and generate the needed portion.

Now after A70 we have terminal "28". That matches needed "28". Then final "27". Good! That matches exactly.

Thus rule_5, with inserted terminals for "30" after first A8, perhaps also need insertion of something else? Let's confirm the full generated string with minimal insertions:

- A2 -> 31 27 A54 A71 A8 32 A8 29 A70 28 27 (this is rule_5)
- A54 -> A55 A14; A55 -> 23; A14 -> 13 A16 (since we need 13); A16 -> 15 (since we need 15) (alternatively there were other possibilities but this matches).
Thus A54 yields "23 13 15".

- A71 expands using choice "A52 26".

A52 expands to: A53 A35 A12 A22.

A53 -> 22 A17 (so yields 22 followed by A17)
A17 -> A18 A19: A18 -> 16; A19 -> 12 A20 (or 13 A21) but we need "12 17". Actually we need after 22 16, the sequence is "12 17". So select A19 -> 12 A20; A20 -> 17 (rule_38). So that yields 12 17.

Then A35 -> 19 A36; we need "19 20". So choose A36 -> 20 (rule_68). So yields 19 20.

A12 -> A13 A14; A13 -> 14, so we get 14 plus A14.

Now we need "13 1". So choose A14 -> 13 A16 (rule_26); then A16 -> 1 (rule_29). So yields "13 1".

Now A22 -> can get 18 via A32->18. So choose A22 -> A32 (via rule_52? Actually rule_52: A22 -> A34? Wait rule 52 is "A22 -> A34". But we need A22 -> A32. Wait let's verify A22 production rules: rule_41: A22 -> A23. rule_42: A22 -> A24. ... rule_50: A22 -> A32. Check rule numbers: rule 49: A22 -> A31. rule_50: A22 -> A32. So yes, rule_50: "A22 -> A32". A32 has productions: rule_62: A32 -> 3; rule_63: A32 -> 18. So we can get "18" by using A22 -> A32, then A32 -> 18. That's two steps but yields terminal 18.

Thus A52 yields the sequence: 22 16 12 17 19 20 14 13 1 18. Perfect.

Then A71 adds a terminal 26 after A52: yields " ... 18 26". So so far we get "22 16 12 17 19 20 14 13 1 18 26". So far matches needed.

- After A71 we have A8. We need a sequence "10 8 11" as the next three symbols. A8 uses rule_17: A8 -> A9 A10. A9 -> 10; A10 -> 8 A11; A11 -> 11 (rule_22). So yields "10 8 11". That's fine.

Now we need "30". But rule_5 has terminal 32 immediately after A8. So we need to insert "30" between the A8 and 32. So we insert a terminal 30 after the first A8 before the existing 32 terminal.

Thus after insertion, we have: ... A8 30 32 A8 ...

- Then we have terminal "32". Good; target has 32 next.

- Next A8 should produce "9". Use rule_16: A8 -> 9.

- Then we have terminal "29" (matches target). Good.

- Next we have A70. We'll choose A70 -> A56 25 (rule_123). Note rule_123: A70 -> A56 25. So we need A56 produce "24 7 21 4 6 5 2 3". Let's check A56's expansion.

A56 -> A57 A50 A3 A37
A57 -> 24 A7 -> 24 7
A50 -> 21 A51 -> 21 4
A3 -> A4 A5 -> 6 (A4) and A5 -> 5 A6 -> 5 then A6-> 2 (choose rule 14). So yields 6 5 2.
A37 -> any of the A3x productions to produce "3". For example, A38 -> 3. So yields "3". So A56 yields: "24 7 21 4 6 5 2 3". That's perfect.

Then A70 adds terminal "25". So after A70, we get "25". Good.

Now after A70 we have terminal "28". That's needed at next position: target includes 28 after 25. Good.

Finally we have terminal "27". Good.

Thus, using rule_5 with just inserted terminal "30" after the first A8, we generate the entire target exactly. Let's verify counts:

Currently we have inserted only one terminal? Wait check if all needed terminals appear in existing rules already: We used terminal "30" inserted. What about terminals "32", "29", "28", "27", etc—they already existed. 30 is present in rule_5? No, rule_5 does not contain 30, but rule_2 and 4 have 30. So we need to insert 30. Are there any other missing terminals? Let's list all tokens in target and confirm they appear somewhere in the grammar when expanding this way.

The target tokens: 31 (present in many rules as starting token; appears in A2 definitions). 27 present as final token in rule_5. 23 from A55; yes. 13 from A14; yes. 15 from A16; yes. 22 from A53; yes. 16 from A18; yes. 12 from A19->12; yes. 17 from A20 -> 17. 19 from A35; yes. 20 from A36 -> 20; yes. 14 from A13; yes. 13 again from A14. 1 from A16 or A20. 18 from A32->18. 26 from A71's appended 26. 10 from A9. 8 from A10. 11 from A11->11. 30 we inserted. 32 from rule_5's explicit terminal. 9 from A8->9. 29 from rule_5's terminal. 24 from A57->24. 7 from A7->7. 21 from A50->21. 4 from A51->4. 6 from A4->6. 5 from A5->5. 2 from A6->2. 3 from A22->A3? Actually A37->A38->3 provides 3. The later 25 is from A70's insertion of 25 after A56; that's part of rule 123. The 25 appears as a terminal in rule A70's production: A70 -> A56 25 (i.e., after A56 we get explicit "25"). So 25 is already present (as part of that alternative). No need to insert. 28 from rule_5's terminal. 27 final from rule_5's terminal.

Thus only missing token is a "30" before the "32". So we need a single insertion.

But we also need to ensure A16 yields 15 for the first part. That is fine, it's there as rule_31: A16 -> 15. So we need to use that production.

Thus the only changes needed are to modify rule_5 to insert terminal "30" after the first A8 (the one after A71). However we must consider that A8 appears twice in that rule: after A71 (first A8), after terminal 32 (second A8). We just need to add "30" after the first A8 and before the 32.

Thus final updated production rule: rule_5 currently: "A2 -> 31 27 A54 A71 A8 32 A8 29 A70 28 27". We need to insert "30" after the first A8.

The syntax could be "A2 -> 31 27 A54 A71 A8 30 32 A8 29 A70 28 27". We'll need to ensure ordering is correct: insert "30" between A8 and 32.

Thus one insertion.

Is there any other needed insertion? Let's try to produce the target using this rule and confirm all tokens align step-by-step:

- Derivation:

Start: A1 -> A2 (rule_1)

A2 -> (rule_5 with inserted "30"): 31 27 A54 A71 A8 30 32 A8 29 A70 28 27

Now expand A54: A54 -> A55 A14 (rule_97)
- A55 -> 23 (rule 98)
- A14 -> 13 A16 (rule 26)
- A16 -> 15 (rule 31)
Thus A54 yields "23 13 15". Good.

Now A71 -> (choose rule_124: A71 -> A52 26). A52 -> A53 A35 A12 A22. Expand:

- A53 -> 22 A17 (rule_96)
- A17 -> A18 A19 (rule_32)
    - A18 -> 16 (rule 33)
    - A19 -> 12 A20 (rule 34) (choose 12 A20)
        - A20 -> 17 (rule 38)
Thus A17 yields 16 12 17.

Thus A53 yields 22 16 12 17.

- A35 -> 19 A36 (rule 66)
    - A36 -> 20 (rule 68)
Thus A35 yields 19 20.

- A12 -> A13 A14 (rule 23)
    - A13 -> 14 (rule 24)
    - A14 -> 13 A16 (again rule 26)
        - A16 -> 1 (rule 29)
Thus A12 yields 14 13 1.

- A22 -> A32 (rule 50)
    - A32 -> 18 (rule 63)
Thus A22 yields 18.

Now A52 yields concatenation: 22 16 12 17 19 20 14 13 1 18

Now A71 yields A52 + 26, i.e., 22 16 12 17 19 20 14 13 1 18 26

Thus far we have: 31 27 [23 13 15] [22 16 12 17 19 20 14 13 1 18 26] then A8 etc.

Now first A8 after A71: Choose rule_17: A8 -> A9 A10. Expand:
- A9 -> 10
- A10 -> 8 A11; choose A11 -> 11 (rule 22)
Thus first A8 yields 10 8 11.

Then insert 30 (our insertion) -> yields terminal 30.

Then terminal 32 part of rule.

Thus we have 10 8 11 30 32

Now second A8 after 32: Use rule_16: A8 -> 9 (to get token 9). So yields 9.

Then terminal 29.

Now A70: choose rule_123: A70 -> A56 25. Expand A56:

- A56 -> A57 A50 A3 A37

* A57 -> 24 A7 (rule_100)
  - A7 -> 7 (rule_15)
=> yields 24 7.

* A50 -> 21 A51 (rule_93)
  - A51 -> 4 (rule_94)
=> yields 21 4

* A3 -> A4 A5. A4->6 (rule 11). A5 -> 5 A6; A6 -> 2 (choose rule 14 to produce 2). So yields 6 5 2.

* A37 -> any of A38- A49: choose A38 -> 3 (rule 81). So yields 3.

Thus A56 yields 24 7 21 4 6 5 2 3.

Then terminal 25 (from A70's production).

Thus after A70, we have "24 7 21 4 6 5 2 3 25". Good.

Then terminal 28 (from rule_5).

Then terminal 27 final.

Thus full generated string:

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

Let's check order matches target sequentially:

Write the sequence:

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

Now compare to target:

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

Exactly matches. Yes, ordering matches perfect.

Thus only one terminal insertion needed.

Now we need to produce output: "Updated production rules:" and list only the changed rule(s). Here only rule_5 changes.

Thus we write:

Updated production rules:
rule_5: A2 -> 31 27 A54 A71 A8 30 32 A8 29 A70 28 27

Should we also note that we inserted the terminal "30" after the first A8. No other changes required.

Thus answer is a single line for updated production rule.

Double-check if any other rules need modifications to ensure that the derivation uses the appropriate productions for A14, A16, A20, A16, A22... But those are already defined; no need to modify. The grammar has multiple productions for some nonterminals (e.g., A14 has options 12 A15 (rule 25) and 13 A16 (rule 26)). We'll pick the appropriate one during parsing; no modifications needed.

Thus minimized insertion is 1 terminal.

We need to ensure no other rule that could conflict because at some points we used A14 -> 13 A16, which is defined. For A16 we used rule_31 (15) for first appearance; and later we used rule_29 (1) for second; that is allowed.

For A16 in the later part (A14 -> 13 A16 after A12 ) we needed A16 -> 1. That's rule_29.

All these options exist, no changes needed.

Thus answer is just the updated rule_5.

Wait: Did we need the A8 after A71 to be rule_17 (i.e., generating 10 8 11)? That is okay.

But note that rule_5's first A8 may expand using rule_17; no insertion needed. However, in rule_5, we inserted a terminal 30 after that A8, but also you might need to consider that rule_5 includes also the token "32". That's correct.

Thus final answer:

Updated production rules:
rule_5: A2 -> 31 27 A54 A71 A8 30 32 A8 29 A70 28 27

Make sure to preserve spaces.

We should not write anything after the updated production rules. That is all.Updated production rules:
rule_5: A2 -> 31 27 A54 A71 A8 30 32 A8 29 A70 28 27