The problem: given a CFG with production rules, we want to insert terminal symbols into the right-hand sides of existing production rules (anywhere) to make the target sequence derivable starting from start symbol A1. No deletion or reordering, no adding new rules. Must minimize total number of inserted terminal symbols.

Goal: Insert missing terminals that were removed. So we need to understand the current grammar and see which terminals appear in target but maybe missing from current derivation possibilities. Also the grammar might be incomplete; need to add terminals at certain positions to make the target sequence derivable.

We must output updated production rules, only listing rules where modifications are made (including rule numbers). Let's parse the given grammar.

Given 24 terminals 1..24, each appears at least once in target. The target sequence:

Indices:

1:24
2:22
3:23
4:7
5:20
6:16
7:10
8:3
9:1
10:14
11:15
12:8
13:2
14:9
15:13
16:21
17:5
18:4
19:6
20:18
21:17
22:11
23:12
24:19

We need to produce this exact sequence via a leftmost derivation (or any derivation order?) Typically to generate a sequence in a CFG, the production rules produce the sequence via successive expansions. The order may be any (but the derivation yields a string). We need to decide about parse tree translation: the target sequence is the concatenation of terminal symbols in the leaves in order. So we must find some derivation from start symbol that yields that exact condition.

Given implementation constraints: we can insert terminals into existing RHSs to fill missing terminals needed.

We need to minimize terminals inserted.

Given that each terminal appears at least once in target, many terminals may be missing from RHS of any rule. Eg: We need to produce terminal 21, 22, 24 etc. Some appear in productions already: e.g., rule_2: A2 -> 24 A28 A30. So we have 24 there. rule_34: A29 -> 22. So 22 is there in A29. The target starts with 24 22; the grammar may produce A2 first produce 24 then something which yields 22 maybe from A28 and A30 expansions.

Let’s analyze the grammar.

List all rules:

1: A1 -> A2

=> start symbol A1 only yields A2. So final expansion must be from A2. So entire target must be derivable from A2. So A2 must derive the target sequence.

2: A2 -> 24 A28 A30

=> starts with terminal 24, then A28, then A30. Good: target starts with 24, then 22, then 23 etc. So we need A28 to generate the rest after 24. But A28 currently leads to A29 via rule_33: A28 -> A29. A29 -> 22 (rule_34). So A28 yields 22. That matches the second token: 22.

After A28 yields maybe just 22 and nothing else. After A28, A30 remains to generate remaining target after 22. So target after 24,22 is 23 7 20 16 10 3 1 14 15 8 2 9 13 21 5 4 6 18 17 11 12 19. So we need A30 -> something that produces this rest.

A30 currently: rule_35: A30 -> A31. So A30 expands to A31.

A31... rule_36: A31 -> A32 A27 A3 A26

Thus A31 yields concatenation: A32, A27, A3, A26.

We need to check if the concatenation of strings from those nonterminals in order can match the rest of target: sequence after 24 22 is ... The rest: 23 7 20 16 10 3 1 14 15 8 2 9 13 21 5 4 6 18 17 11 12 19.

Thus we need A32 produce prefix 23 maybe? Let's see A32: rule_37: A32 -> 23 A7

So A32 yields terminal 23 then whatever A7 yields. A7 -> 7 (rule_8). So A7 yields 7. So A32 yields "23 7". That matches first two of the rest: indeed after 24 22 we have 23 7. Good.

After A32, we have A27, A3, A26.

Let's see what A27 currently: rule_32: A27 -> 20 A22.

Thus A27 yields terminal 20 then strings from A22.

A22 rule: A22 -> A23 A21 A8 A20 (rule_27). So we need to produce: after "23 7", we need the remainder: 20 16 10 3 1 14 15 8 2 9 13 21 5 4 6 18 17 11 12 19. So we need A27 to yield 20 then something equal to start of remainder: "16..." So A27 = 20 + A22. That matches 20 after 23 7.

Now A22 -> A23 A21 A8 A20. So A22 yields concatenation of A23, A21, A8, A20.

Let's check each:

- A23: rule_28: A23 -> 16 A13

So A23 yields terminal 16 then A13.

A13 rule: ??? rule_16: A13 -> A14 A15

Thus A13 yields A14 then A15.

A14 -> 10 (rule_17). So gives 10.

A15 has two rules: rule_18: A15 -> 2 A17; rule_19: A15 -> 3 A16.

Thus A15 can produce either "2 A17" or "3 A16". For our target we need "10 3 1" next after "16 10"? Let's see target after 20: It's "16 10 3 1". Yes we have 16 (via A23), 10 (via A14), then target expects "3 1". Therefore we need to choose A15 -> 3 A16. That yields 3 then A16. A16 -> 1 (rule_20). So A15 yields "3 1". Good. So A23 yields "16 10 3 1". That matches target sequence after the 20, up to then we have at next token after that? Let's list: After 20, we have 16 10 3 1. So far good.

Now after A23, A22 yields A21. What is A21? rule_26: A21 -> 14 15 (two terminals). So A21 yields "14 15". That matches the next two tokens in target: we have after "16 10 3 1" token sequence: "14 15". Good.

Now after A21, we have A8. A8 -> A9 A10 (rule_9). A9 -> 8 (rule_10). So that yields 8. A10 has two alternatives: rule_11: A10 -> 2 A11; rule_12: A10 -> 3 A12. Target after "14 15" is "8 2 9". So we can use A10 -> 2 A11 to generate 2 followed by A11. A11 rule: rule_13: A11 -> 1; rule_14: A11 -> 9. Terminal needed: target after 2 is 9 (since we've got 8 2 9). Thus we need A11 -> 9. So we choose A10 -> 2 A11 => 2 9. So the A8 yields 8 + (2 9) = 8 2 9. Good.

Now after A8, we have A20. A20 -> 13 (rule_25). Target after (8 2 9) is "13". Good.

So A22 yields: A23 (16 10 3 1) + A21 (14 15) + A8 (8 2 9) + A20 (13). This yields "16 10 3 1 14 15 8 2 9 13". Indeed after 20, we have exactly that. So far everything matches target until after that we have token "21". Indeed target after "13" is "21". So we still have A3 and A26 to produce the tail: "21 5 4 6 18 17 11 12 19". Let's see.

A3 rule: A3 -> A4 A5 (rule_3).

A4 -> 5 (rule_4). So yields 5.

A5 -> 4 A6 (rule_5). So yields 4 and then A6. A6 has two productions: rule_6: A6 -> 1; rule_7: A6 -> 6. The target after "13" we currently have "21 5 4 6 ...". The next tokens in sequence: "21". So we cannot produce 21 from A3, maybe A26 yields it? Let's see.

A26 rule: A26 -> 18 A24 (rule_31).

A24 -> A25 A18 (rule_29). A25 -> 17 (rule_30). A18 -> 11 A19 (rule_22). A19 -> 1 (rule_23) or A19 -> 12 (rule_24). Wait we have some ambiguous possibilities.

We also need terminal "21" somewhere presumably. I don’t see any rule that yields 21 as a terminal. The only terminal 21 is not present in any of the RHS currently. Let's search: Is there any rule using terminal 21? No, I read all rule RHSs: look at each: rule_2: 24; rule_4: 5; rule_5: 4; rule_6: 1; rule_7:6; rule_8:7; rule_10:8; rule_11:2; rule_12:3; rule_13:1; rule_14:9; rule_15:1; rule_17:10; rule_18:2; rule_19:3; rule_20:1; rule_21:1; rule_22:11; rule_23:1; rule_24:12; rule_25:13; rule_26:14 15; rule_28:16; rule_30:17; rule_31:18; rule_32:20; rule_34:22; rule_37:23; rule_8:7; rule_3 (A3->A4 A5) no terminals; rule_9 (no terminals); etc.

Thus terminal 21 appears nowhere. That's a problem because the target includes 21.

Given restrictions, we can only insert terminals into existing RHSs; we cannot create a new rule. So we must insert a "21" somewhere within RHS of existing rules, such that it is part of the derived string.

Since we need exactly one "21" at position after token 13, before token 5 (the production A4). Actually target sequence: After 13, token order is: 21 5 4 6 18 17 11 12 19.

Our derivation currently after processing A22, we have processed A3 (which gives "5 4 ?") and A26 (which yields something like "18 17 11 ..." etc). But we lack the "21". Perhaps we can insert "21" somewhere in a rule that yields a positioning between A3 and A26, i.e., maybe in A31's RHS we have A32 A27 A3 A26. The order is A32 (generates 23 7), A27 (generates 20 and A22 ...), A3 (generates 5 4 A6...), A26 (generates 18 ..., etc). So after A3 we have A26. The "21" appears between 13 (end of A22) and 5 (start of A3). So if A3 currently yields "5 4 ?" but we need "21" before the "5". So we could insert "21" at the beginning of A3's RHS: e.g., modify rule_3: A3 -> 21 A4 A5. Since rule 3 originally is "A3 -> A4 A5". Inserting 21 before A4 gives A3 -> 21 A4 A5. That would cause A3 to produce 21 then A4 (5) then A5 (4 A6). That would put 21 before 5, which matches the desired order: 21 5 4 ... However, we must also consider that A6 currently can produce either 1 or 6. But we need after 4, we need 6 (target token after 4 is 6). Indeed target after "5 4" is "6". A6 currently can produce 1 (rule_6) or 6 (rule_7). We need 6. So choose A6 -> 6. Fine. So after A3 we produce "21 5 4 6". That covers tokens 21 5 4 6.

After A3 we have A26 (prev). A26 yields "18 A24". A24 -> A25 A18. A25 -> 17. A18 -> 11 A19. A19 -> either 1 or 12. To match remaining target after 21 5 4 6, we have tokens: 18 17 11 12 19. Let's see if we can produce that. A26 gives 18, then A24 yields A25 A18 = 17 (from A25) and A18 => 11 A19. So after 17 we have "11" then A19. A19 can yield 1 (rule_23) or 12 (rule_24). We need token 12 after 11. Actually target after 11 is 12. So we need A19 -> 12 (choose rule_24). That yields token 12. So A26 yields "18 17 11 12". That's four tokens. Then final token "19". There is no rule that yields terminal 19. Indeed we have no rule with RHS containing "19". So need to insert "19" somewhere in the derivation.

Potentially we have A19 being used to produce maybe 1 or 12. Could we insert 19 after A19? Maybe modify A19's productions to include a 19 after the terminal (like A19 -> 1 19). That would produce "1 19" or "12 19". But we need only "19" after "12". However here we have no 1 needed after "12". So we could modify rule_24: A19 -> 12 19. That would yield "12 19". Then A18 would produce "11" then A19 yields "12 19". That gives "11 12 19". Perfect.

Alternatively we could modify rule_23: A19 -> 1 19…but we need "1"? Not needed. So modify rule_24.

Now we need to consider the missing 21 we inserted into A3. We need to insert that one terminal.

Now we need to also ensure "1" after "3"? Wait we have used A6 ->6 and not used "1". Did we need a "1" anywhere after "4"? Let's examine the target: after "21 5 4", we have "6". Yes after "4", we have "6". So we choose A6->6, correct.

Now check all tokens after "13". The remaining target tokens are:

14:21? Let's list again:

Indices:

1:24
2:22
3:23
4:7
5:20
6:16
7:10
8:3
9:1
10:14
11:15
12:8
13:2
14:9
15:13
16:21
17:5
18:4
19:6
20:18
21:17
22:11
23:12
24:19

We have derived:

A1 -> A2 -> 24 A28 A30

A28 -> A29 -> 22

A30 -> A31 -> A32 A27 A3 A26

A32 -> 23 A7 -> 23 7

A27 -> 20 A22 -> 20 then A22

A22 -> A23 A21 A8 A20

A23 -> 16 A13 -> 16 then A13

A13 -> A14 A15 -> 10 then A15

A15 choose rule_19 -> 3 A16 -> produce 3 then A16 -> 1 (but we need 3 1). target after 10 is 3 1, correct.

Thus A23 yields "16 10 3 1". Wait we need to verify: A23 -> 16 A13; A13 -> A14 A15 = 10 + (A15). A15 -> 3 A16 = "3" + A16. A16 -> 1. So yes A23 yields "16 10 3 1". Good.

Then A21 yields "14 15". Good.

Then A8 yields 8 then A10 => 2 A11 -> 2 + A11 (choose "9"). So overall yields "8 2 9". Good.

A20 -> 13.

Thus A22 yields "16 10 3 1 14 15 8 2 9 13". This matches tokens after "20". Indeed after "20", we had "16 10 3 1 14 15 8 2 9 13". Yes.

Now A3 should yield "21 5 4 6". After injection, rule_3 will be "A3 -> 21 A4 A5". A4 yields 5, A5 yields 4 A6. A6 -> 6 (choose rule_7). So A3 yields "21 5 4 6". Good.

Now A26 yields "18" + A24. A24 -> A25 A18 -> 17 A18. A18 -> 11 A19. A19 -> 12 19 (modified). So A26 yields "18 17 11 12 19". Good.

Thus the whole derived string concatenated: 

- 24
- 22
- 23 7
- 20
- 16 10 3 1 14 15 8 2 9 13
- 21 5 4 6
- 18 17 11 12 19

Combine:

24 22 23 7 20 16 10 3 1 14 15 8 2 9 13 21 5 4 6 18 17 11 12 19

Exactly matches target. Great.

Thus required modifications: Insert terminal 21 at beginning of rule_3 RHS (after arrow). Insert terminal 19 after terminal 12 in rule_24 RHS (so A19 -> 12 19). Also we need to modify rule_7? Actually A6 had two productions: rule_6 (-> 1) and rule_7 (->6). We want to produce 6, we already have rule_7 as 6, which is fine. No changes needed.

Thus modifications: rule_3, rule_24.

We need to output updated production rules lines only for those where changes were made, in the same format as given (e.g., "rule_3: A3 -> 21 A4 A5").

Now check if any other modifications needed: Possibly for A3 we inserted 21 before A4 A5. But does the phrase "anywhere in existing production rules" allow insertion before first non-terminal? Yes. The rule originally had "A3 -> A4 A5". So we can insert a terminal anywhere: before A4, after A4, between A4 and A5, after A5. We inserted at start: "21 A4 A5". That's permissible.

For rule_24, originally "A19 -> 12". Inserting after 12 we get "12 19". That's also allowed.

Check if any other missing terminals: all tokens are covered. Let's double-check presence of each terminal from 1 to 24.

List by occurrence from grammar after modifications:

- 1: appears in A6 ->1 in rule_6, also in A15 ->3 A16 -> A16->1; also in ... but after insertion we use A6->6 not 1; still some 1 appears earlier (A15->3 A16 -> 1). So we have at least one instance.

- 2: appears in A10 ->2 A11.

- 3: appears in A15 ->3 A16, also A10 ->3 A12 (unused maybe), produce later.

- 4: appears in A5 ->4 A6.

- 5: appears in A4 ->5.

- 6: appears in A6 ->6 (unused originally? Actually rule_7 is "A6 -> 6". Yes.

- 7: in A7 ->7.

- 8: in A9 ->8.

- 9: A11 ->9.

- 10: A14 ->10.

- 11: A18 ->11 A19.

- 12: now A19 ->12 19. So 12 appears.

- 13: A20 ->13.

- 14: A21 ->14 15: first terminal 14.

- 15: A21 ->14 15: second terminal 15.

- 16: A23 ->16 A13.

- 17: A25 ->17.

- 18: A26 ->18 A24.

- 19: inserted in A19 ->12 19.

- 20: A27 ->20 A22.

- 21: inserted in A3 ->21 A4 A5.

- 22: A29 ->22.

- 23: A32 ->23 A7.

- 24: A2 ->24 A28 A30.

Thus all 1-24 appear in some production rules. So constraints satisfied.

Now check that modifications do not break other derivations. Our chosen derivation used rule_3 as "21 A4 A5". We might need to ensure no ambiguity from having multiple productions for A3? There's only this rule (original rule_3). The grammar allowed only one rule for each LHS. So after modification, the rule becomes "A3 -> 21 A4 A5". That yields exactly our needed output.

Now we need to output:

Updated production rules:

rule_3: A3 -> 21 A4 A5

rule_24: A19 -> 12 19

Now we must ensure proper formatting: We need to include the rule numbers. Should we also include the old productions unchanged? No, they said "no need to rewrite the production rules which haven't changed." So only changed.

Thus final answer.

You might verify if any overlapping duplicate terminals cause ambiguous parse? It's okay.

Thus answer ready.Updated production rules:
rule_3: A3 -> 21 A4 A5
rule_24: A19 -> 12 19