We have a CFG grammar given. Several terminals possibly missing due to removal, causing inability to derive target sequence. We need to insert terminal symbols into existing productions to enable derivation, with minimal insertions. Only allow insertions anywhere in existing rules, not reorder or delete symbols, nor add new production rules. So we can add terminals within RHS strings (before or after symbols, or between non-terminals?) Yes. Must produce target sequence: "17 11 2 1 15 4 12 3 13 14 19 20 21 9 8 10 23 5 7 6 22 16 18". Let's list them with indices:

Target: (probably a sequence of 23 terminals, each integer from 1-23 must appear at least once; indeed sequence includes 1,2,... maybe missing some earlier? Let's list: 1 appears at positions after 2 etc. Let me list in order:

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

All numbers from 1 to 23 present exactly once? Let's check: we have 1,2,3,... up to 23. Indeed... The sequence is a permutation of 1..23: 1 occurs at position 4; 2 at pos3; 3 at pos8; 4 pos6; 5 pos18; 6 pos20; 7 pos19; 8 pos15; 9 pos14; 10 pos16; 11 pos2; 12 pos7; 13 pos9; 14 pos10; 15 pos5; 16 pos22; 17 pos1; 18 pos23; 19 pos11; 20 pos12; 21 pos13; 22 pos21; 23 pos17.

The CFG seems designed to produce these in some order maybe concatenation of sub-structures. Our job is to insert missing terminals.

The given grammar has starting symbol A1 -> A2. A2 makes A29 20 A30 A28. So start yields concatenation: A29, terminal 20, A30, A28.

Thus sequence produced must start with whatever A29 expands to, then 20, then A30 expands to some sub-sequence, then A28 expands.

Goal: produce exact target sequence left-to-right: start with 17, 11, ..., 18. So let's try to parse target: A29 should generate 17 11 ... something before encountering terminal 20? According to rule_2: A2 -> A29 20 A30 A28. So after A29's expansion, we get terminal 20. In target, the term 20 occurs at position 12 (value 20). So A29 must generate the prefix of target up to before that terminal at position 12. Let's compute that: target prefix before 20 is up to 19: positions 1-11 = 17 11 2 1 15 4 12 3 13 14 19. Then we have 20 at position 12, matches rule_2's terminal. Good. So A29 must produce sequence: 17 11 2 1 15 4 12 3 13 14 19. And after 20, A30 must produce the suffix after 20 and before A28 later. After 20, target suffix is from position13 onward: 21 9 8 10 23 5 7 6 22 16 18. Then A28 must produce what remains after A30? We'll need to parse that later.

A28 has rule_36: A28 -> A26 18. So A28 yields expansion of A26, then terminal 18 at end. Target ends with 18 at position 23. So this matches that A28's last symbol is 18, as required. So A26 must generate the sequence before the final 18, i.e., target prefix after position 13? Wait, A30 comes before A28; after A30 expansion the next is A28 expansion. So entire sequence after 20 is: A30 expansion followed by A26 expansion then 18. So target suffix after 20 is: (positions 13-23) = 21 9 8 10 23 5 7 6 22 16 18. So after we produce target after 20, final 18 must arise from A28's terminal. So A28's A26 must generate 21 9 8 10 23 5 7 6 22 16 (positions 13-22). So we have A26 generating that.

Thus need to derive A29 -> prefix 1..11, and A30 -> some part of target positions maybe a prefix of the 21... sequence, and A26 -> remainder before final 18 (positions 13-22). Also A30 may produce some more elements including maybe terminal 21 etc.

Now we need to see the current definitions for A29 and A30, and see how they can be expanded through existing productions (with possible insertions). Also need to derive A26.

List existing productions:

- rule_1: A1 -> A2
- rule_2: A2 -> A29 20 A30 A28
- rule_3: A3 -> 2 A4
- rule_4: A3 -> 3 A5
- rule_5: A4 -> 1
- rule_6: A5 -> 1
- rule_7: A6 -> A7 A8
- rule_8: A7 -> 5
- rule_9: A8 -> 7 A9
- rule_10: A8 -> 8 A10
- rule_11: A9 -> 1
- rule_12: A9 -> 6
- rule_13: A10 -> 1
- rule_14: A11 -> A12 A13
- rule_15: A12 -> 9
- rule_16: A13 -> 7 A14
- rule_17: A13 -> 8 A15
- rule_18: A14 -> 1
- rule_19: A15 -> 1
- rule_20: A15 -> 10
- rule_21: A16 -> A17 A3
- rule_22: A17 -> 11
- rule_23: A18 -> A19 A20
- rule_24: A19 -> 12
- rule_25: A20 -> 2 A22
- rule_26: A20 -> 3 A21
- rule_27: A21 -> 1
- rule_28: A21 -> 13
- rule_29: A22 -> 1
- rule_30: A23 -> 14
- rule_31: A24 -> 15 A25
- rule_32: A25 -> 4
- rule_33: A26 -> 16
- rule_34: A26 -> A27 A24 A18 A23
- rule_35: A27 -> 17 A16
- rule_36: A28 -> A26 18
- rule_37: A29 -> A26 19
- rule_38: A30 -> 21 A11 A6

Alright. Observations:

- A29 can produce either (via rule_37) A26 19. Currently A26 produces 16 (or via longer production rule_34). Then A29 following that has terminal 19 at end. But our target for A29 should be: 17 11 2 1 15 4 12 3 13 14 19. Let's parse: A29's target ends with 19. Good: rule_37 says nTerminal 19 from A29 at end. The preceding part A26 must generate 17 11 2 1 15 4 12 3 13 14? Actually note that after A26 expansion, then terminal 19 yields the final 19. So A26 must produce sequence up to 14 (position 10). But also note that the target's 19 is at position 11. So A26 must generate 17 11 2 1 15 4 12 3 13 14 (positions 1-10). Let's confirm: positions 1-10 are: 17,11,2,1,15,4,12,3,13,14. Good.

So we need to be able to derive A26 -> those 10 terminals, using the existing rules with insertions only.

A26 has two productions:

- rule_33: A26 -> 16

- rule_34: A26 -> A27 A24 A18 A23

Thus we can select A26 -> A27 A24 A18 A23 (since we need many terminals!). A27 -> 17 A16 (rule_35). A16 -> A17 A3 (rule_21). A17 -> 11 (rule_22). So A27 -> 17 A16, then A16 expands to A17 A3, so far we have 17 11 A3. Good: matches prefix 17 11.

Now we need after A3 the rest of the prefix: 2 1 15 4 12 3 13 14.

A3 can produce either 2 A4 (rule_3) or 3 A5 (rule_4). A4 -> 1 (rule_5). A5 -> 1 (rule_6). So options:

- A3 -> 2 A4 -> 2 1
- A3 -> 3 A5 -> 3 1

Thus we can produce either "2 1" or "3 1". In target after 17,11 we have "2 1". Good, we can use the first production to get "2 1". So we need A3 -> 2 A4 (no insertion needed). Great.

So A27 A24 A18 A23 expansion yields:

A27: 17 A16

- A16: A17 A3 -> 11 followed by 2 A4 -> 2 1.

Thus far: 17, 11, 2, 1.

Now we need to produce rest "15 4 12 3 13 14". Continue after A16 we have A24 then A18 then A23.

Let's look at A24: rule_31: A24 -> 15 A25. Great: 15 then A25.

A25: rule_32: A25 -> 4. So A24 expands to 15 4. Good! That yields "15 4" after current prefix. So now sequence is 17 11 2 1 15 4, matches target positions 1-6.

Now A18: rule_23: A18 -> A19 A20. So we need to produce "12 3 13 14". Let's see:

A19 -> 12 (rule_24). Good. So then A20 must generate "3 13 14". A20 has two productions:

- rule_25: A20 -> 2 A22
- rule_26: A20 -> 3 A21

Our needed sequence after 12 is "3 13 14". So we can choose A20 -> 3 A21, matching leading terminal 3. Good.

Now A21 can produce either 1 (rule_27) or 13 (rule_28). We need after 3 "13 14": i.e., the terminal after 3 is 13. So we choose A21 -> 13 (no extra insertion). That yields 13.

Now after A21 we have to produce "14". A23 is after A18: A18's expansion is A19 A20 (just those). After that, the next symbol is A23 (from A26's expansion). A23 is defined via rule_30: A23 -> 14. Perfect! So A23 yields terminal 14. So A24 A18 A23 yields "15 4 12 3 13 14". Combined with earlier prefix, we have exactly "17 11 2 1 15 4 12 3 13 14". Great! That's exactly the 10 terminals we needed before the final 19. So A26 → A27 A24 A18 A23 yields "17 11 2 1 15 4 12 3 13 14". Then A26 followed by terminal 19 (in rule_37) yields A29 → "17 11 2 1 15 4 12 3 13 14 19". So A29 yields correct prefix before 20. Thus A29 -> A26 19 yields exactly the needed sequence up to position 11 (including 19). No insertions needed! Good!

Now consider after 20, we have A30 (from rule_2). Rule_38: A30 -> 21 A11 A6.

Thus A30 yields terminal 21 then A11 then A6.

Goal after 20 in target is: positions 13-22 (excluding final 18) = 21 9 8 10 23 5 7 6 22 16 (and then final 18 from A28). Wait our earlier segmentation: After 20, suffix up to before final 18 (which is part of A28) is: 21 9 8 10 23 5 7 6 22 16  (positions 13-22). So A30 must produce the initial part among these, while A26 (in A28) must generate remaining 9,... Actually A30 is before A28; A28 -> A26 18. So A30 provides prefix of the suffix; A26 (within A28) provides some middle part before 18. So combined A30 + A26 must match the sequence 21 9 8 10 23 5 7 6 22 16. Then final 18 as terminal.

Thus we need to split this stretch between A30 and A26.

A30 = 21 A11 A6.

Thus first terminal of this part is 21, matches target's first term after 20: indeed position13 = 21. Good.

After 21, we need to produce "9 8 10 23 5 7 6 22 16" using A11 A6 expansions.

Let's explore definitions:

A11 -> A12 A13 (rule_14). A12 -> 9 (rule_15). So A12 yields terminal 9. Good. A13 has two productions:

- rule_16: A13 -> 7 A14
- rule_17: A13 -> 8 A15

Additionally, A14 -> 1 (rule_18). A15 -> 1 (rule_19) and also A15 -> 10 (rule_20). So A15 can produce either 1 or 10 depending on production used.

Now A6 -> A7 A8 (rule_7). A7 ->5 (rule_8). A8 has two productions:

- rule_9: A8 -> 7 A9
- rule_10: A8 -> 8 A10

A9 -> 1 (rule_11) or 6 (rule_12). A10 ->1 (rule_13).

Thus we need to derive the rest of the sequence "9 8 10 23 5 7 6 22 16". Let's outline possible expansions.

- A11 yields A12 A13 = 9 followed by A13 expansion.

Then A6 yields A7 A8 = 5 followed by some from A8.

Thus overall A30 yields 21, then A11 part: 9 + something, then A6 part: 5 + something.

Goal after 21: "9 8 10 23 5 7 6 22 16". Let's break:

- After 21, we need 9 (fits A12). Good.

- Then we need "8 10 23 5 7 6 22 16". That could be the rest (A13 + A6). But we have 5 after A6 anyway.

Better to examine possible sequences from A13 and A6.

A13 can be either "7 1"(with A14->1) or "8 1" (A15->1) or "8 10" (A15->10). Wait A13 productions: A13 -> 7 A14 (A14->1). So yields "7 1". A13 -> 8 A15. A15 can produce 1 or 10. So yields either "8 1" or "8 10". So these expansions produce sequences: "7 1", "8 1", "8 10". After that we have A6.

A6: A7 A8. A7 is 5. A8 can be either "7 A9" (where A9 yields 1 or 6) or "8 A10" (A10 yields 1). So options:

- A8 -> 7 A9 -> 7 (1 or 6) yields "7 1" or "7 6".
- A8 -> 8 A10 -> 8 1 yields "8 1".

Thus A6 yields 5 followed by either "7 1", "7 6", or "8 1".

Thus total after A30: 21 [A12=9] A13 then 5 [A7] then one of those combos.

Goal sequence: 9 8 10 23 5 7 6 22 16.

Observe after 21 we have 9, then 8, then 10, then 23, then 5, then 7, then 6, then 22, then 16. So suffix after 21 is: [9] [8 10] [23] [5 7 6] [22 16]. Let's parse: A12 = 9. A13 could be "8 10". Perfect. Then we need "23". Where does 23 come from? There is no production that directly outputs 23 except perhaps via some nonterminal we haven't seen, but A23 defined earlier is 14. Not 23. Do we have any other productions with terminal 23? No. Let's search through given productions for any 23: None? Let's scroll. There's a terminal 23 in target, but no rule explicitly with 23? Let's inspect all rules:

- rule_1: A1 -> A2 (no terminals)
- rule_2: A2 -> A29 20 A30 A28 (20)
- rule_3: A3 -> 2 A4 (2)
- rule_4: A3 -> 3 A5 (3)
- rule_5: A4 -> 1
- rule_6: A5 -> 1
- rule_7: A6 -> A7 A8
- rule_8: A7 -> 5
- rule_9: A8 -> 7 A9
- rule_10: A8 -> 8 A10
- rule_11: A9 -> 1
- rule_12: A9 -> 6
- rule_13: A10 -> 1
- rule_14: A11 -> A12 A13
- rule_15: A12 -> 9
- rule_16: A13 -> 7 A14
- rule_17: A13 -> 8 A15
- rule_18: A14 -> 1
- rule_19: A15 -> 1
- rule_20: A15 -> 10
- rule_21: A16 -> A17 A3
- rule_22: A17 -> 11
- rule_23: A18 -> A19 A20
- rule_24: A19 -> 12
- rule_25: A20 -> 2 A22
- rule_26: A20 -> 3 A21
- rule_27: A21 -> 1
- rule_28: A21 -> 13
- rule_29: A22 -> 1
- rule_30: A23 -> 14
- rule_31: A24 -> 15 A25
- rule_32: A25 -> 4
- rule_33: A26 -> 16
- rule_34: A26 -> A27 A24 A18 A23
- rule_35: A27 -> 17 A16
- rule_36: A28 -> A26 18
- rule_37: A29 -> A26 19
- rule_38: A30 -> 21 A11 A6

No explicit 23 anywhere. That's a problem: we need to produce terminal 23, but there is no production that yields it. However, we can insert terminals into existing productions. Since we can insert missing terminals anywhere in RHS, we can insert "23" at appropriate location in a rule's RHS. It should be inserted into some existing rule such that the yield of that rule includes the terminal. Goal: generate 23 after the 10 after A13. The sequence after A13's expansion (currently "8 10" if we choose that) yields "8 10". Then we need to produce "23". We could insert "23" after the "10" in the production that yields "10". That's rule_20: A15 -> 10. Currently this production yields a single terminal 10. If we insert a terminal after 10 (e.g., "10 23") then A15's expansion would produce "10 23". But we need "10" to be part of A13's expansion. A13 -> 8 A15. So currently with A15 -> 10, A13 yields "8 10". If we change rule_20 to "10 23", then A13 yields "8 10 23". That's exactly what we need: 8 10 23. Good.

But we must ensure that adding "23" doesn't break other usages of A15 that rely on just a terminal "10". However A15 also appears as A13 -> 8 A15. That's the only occurrence of A15? Let's search: A15 appears only in rule_17 (A13 -> 8 A15) and rule_20 and rule_19 (A15 -> 1). So if we add a terminal after 10 in rule_20, A15 could produce "10 23". Similarly A15's other production rule_19: A15 -> 1, could optionally be used elsewhere if we needed. Changing rule_20 is safe: it only affects the branch where we chose rule_20. In our derivation for target we need A15->10 (the variant that yields 10). After insertion, it yields "10 23". That yields the needed "23". Good.

Now after that we need "5 7 6". That must be generated by A6: A7 A8. As we said, A7 is 5, fine. A8 must produce sequence "7 6". There are two productions: A8 -> 7 A9 (then A9 -> 6 yields "7 6") or A8->7 A9 (with A9->1 yields "7 1").

Thus we can choose A8 -> 7 A9, and A9 -> 6. That yields "7 6". So A6 yields 5, then 7, then 6. That's exactly needed sequence.

Now after that we have "22 16". This must be generated by the later A26 in A28.

Recall A28 -> A26 18. So A26 must generate "22 16" (positions 21-22 before final 18). The target's positions 21-22 are 22 and 16 in order. And we have A26's possible productions: either terminal 16 (rule_33) or A27 A24 A18 A23. The latter produces a long prefix we used earlier, but maybe we could also derive "22 16". However we might need to produce both 22 and 16. Let's examine possibilities.

Option 1: Use rule_33 A26 -> 16 to produce terminal 16, but then we need also a preceding 22. Where could 22 be generated? Perhaps we can insert 22 in the rule for A26 with longer expansion? Perhaps at appropriate spot we could insert 22 somewhere else. Or use alternative nonterminal to produce 22 before A26.

But note that after A6 we have A28's A26 (the one on the other side) to produce the remaining part. Usually we expected A26 -> ... and then 18. Actually we have two A26 symbols in the grammar: one on the left side of rule_36: A28 -> A26 18. And another earlier inside A29's production rule_37: A29 -> A26 19 (the same A26). So A26 is used in both contexts: A26 inside A29 we used for big representation; and A26 in A28 we need to use for suffix region.

But if we use A26's longer production for both, A29's and A28's expansions would be identical, which is not what we want because A29's A26 yields the prefix of length 10; but A28's A26 must produce length 2 (22 and 16). However note: Grammar allows same nonterminal symbol A26 to derive both ways (sharing same productions). We can have two different derivations using different productions for each occurrence. That's fine. So for A26 inside A28, we could choose a different production: e.g., A26 -> 16 (rule_33), generating just 16. Then preceding 22 must be generated somewhere else: perhaps we can insert 22 before 16 in rule_33: change rule_33 to "16" currently. But we cannot reorder, we can only insert terminals anywhere. So we could change rule_33 to "22 16" or "16 22"? Let's consider.

If we change rule_33 to "22 16", then A26 would generate "22 16". But recall that A26 is also used inside A29, where we earlier used the long production A27 A24 A18 A23 exactly to generate 10 terminals. However, note that A26 also has the production A26 -> 16 (rule_33). But in A29 derivation we used different production (A34: A26 -> A27 A24 A18 A23). So we can keep rule_33 as changed to "22 16". That won't affect A29 because we used the longer production for that occurrence. Great.

Thus we can modify rule_33 to "22 16". Wait but the idea is we need A26 to generate 22 followed by 16. Since rule_33 originally "16", we could insert "22" before the 16: "22 16". That yields correct order. This is an insertion and maintains order: we are not deleting or reordering. So rule_33 becomes: A26 -> 22 16. That adds a terminal 22. That is one insertion.

We must also consider that the original grammar has other productions using A26 to generate 16 alone? There is also A26 -> A27 A24 A18 A23 (rule_34). That remains unchanged. So A26 can generate either 22 16 (modified rule_33) or longer prefix. That's fine.

Thus after A6 we have A28 -> A26 18, where A26 (via rule_33) yields "22 16". Wait but order is "22 16"? Actually target's order is "22 16" before 18. We need "22 16". Yes.

Thus A28 yields "22 16 18". Perfect.

Thus modifications needed:

- Insert terminal 23 after terminal 10 in rule_20: change rule_20 from "A15 -> 10" to "A15 -> 10 23". That's one insertion.

- Insert terminal 22 before terminal 16 in rule_33: change rule_33 from "A26 -> 16" to "A26 -> 22 16". That's another insertion.

Thus total insertions = 2. Let's double-check that everything else yields correct ordering.

Full derivation:

Start: A1 -> A2

A2 -> A29 20 A30 A28

A29 -> A26 19 (using rule_34 for A26)
   (expand A26 via rule_34 -> A27 A24 A18 A23)
      A27 -> 17 A16
        A16 -> A17 A3 (since need 11 and then 2 1)
           A17 -> 11
           A3 -> 2 A4 (choose rule_3)
               A4 -> 1
         So A27 expands to 17 11 2 1
      A24 -> 15 A25 (rule_31)
         A25 -> 4 (rule_32)
      So A24 expands to 15 4
      A18 -> A19 A20
         A19 -> 12
         A20 -> 3 A21 (rule_26)
            A21 -> 13 (rule_28)
      So A18 expands to 12 3 13
      A23 -> 14
   So A26 (via rule_34) yields: 17 11 2 1 15 4 12 3 13 14
   Then A29 adds terminal 19 => 17 11 2 1 15 4 12 3 13 14 19

Thus A2 yields prefix of target exactly up to 19.

Now in A2, after A29 we have terminal 20. Then A30.

A30 -> 21 A11 A6

A11 -> A12 A13
   A12 -> 9
   A13 -> 8 A15 (rule_17)
       A15 -> 10 23 (modified rule_20)
 So A13 yields 8 10 23
 So A11 yields 9 8 10 23

A6 -> A7 A8
   A7 ->5
   A8 -> 7 A9 (rule_9)
      A9 -> 6 (rule_12)   (alternatively we could output 1 but we need 6)
 So A6 yields 5 7 6

Thus A30 yields: 21 9 8 10 23 5 7 6.

That matches target "21 9 8 10 23 5 7 6". Good.

Now after A30 we have A28.

A28 -> A26 18

We need A26 to output 22 16 (inserted). We use modified rule_33 "A26 -> 22 16". Then A28 yields 22 16 18.

Thus final full sequence:

A1->...-> A29 (prefix) => 17 11 2 1 15 4 12 3 13 14 19
then terminal 20 => 20
then A30 => 21 9 8 10 23 5 7 6
then A28 => 22 16 18

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

Matches target exactly: [target earlier]. Indeed check you have the same sequence:

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

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

Thus only needed insertions: add 23 to rule_20 after 10; add 22 before 16 in rule_33.

We need to output "Updated production rules:" and each updated rule with rule_<n>:. So we need to list rule_20 and rule_33 with inserted terminals.

But need to be careful about format: "rule_20: A15 -> 10 23" and "rule_33: A26 -> 22 16". Actually original rule_20 is "A15 -> 10". Inserted "23". Insert after the terminal. So becomes "A15 -> 10 23". Good.

And rule_33 is "A26 -> 16". Insert "22" before 16: "A26 -> 22 16". Valid.

Make sure we not modify any other rule.

Now, are there any other potential missing terminals? Let's double-check each terminal appears exactly once in target. We have used all. Check terminals from 1 to 23:

- 1: appears multiple times in grammar (anywhere?). In target appears at position 4. It appears as part of A3->2 A4, etc. There's also some 1s in other productions but they may not be used in actual derivation. That's fine.

- 2: target after 11 there is 2 (position3). We used A3->2 A4.

- 3: target position8 (value 3). We used A20->3 A21.

- 4: target pos6 (value 4). We used A25->4.

- 5: target pos18 (value5). A7->5.

- 6: target pos20 (value6). A9->6.

- 7: target pos19 (value7). A8->7 A9.

- 8: target pos15 (value8). A8->8 A10? Actually we used A13->8 10 (in A11). That's also part of target pos15? Wait target has 8 at position15. Our derived sequence had "8" as part of A13? Yes, the sequence "9 8 10 23" we have 8 at position15 (correct). We also have "5 7 6", but there is also "8"? There's also "8" before 10 in target after 9. Yes.

- 9: target pos14 is 9. A12 -> 9.

- 10: target pos16 is 10. A15 -> 10.

- 11: target pos2 is 11. A17 -> 11.

- 12: pos7. A19 -> 12.

- 13: pos9. A21->13.

- 14: pos10. A23->14.

- 15: pos5. A24->15.

- 16: pos22. A26->16 (the terminal 16 we inserted). Good.

- 17: pos1. A27->17.

- 18: pos23. A28's terminal 18.

- 19: pos11. A29's terminal 19.

- 20: pos12. A2's terminal 20.

- 21: pos13. A30's terminal 21.

- 22: pos21. inserted in A26's production.

- 23: pos17 inserted after 10.

Thus all present. Good.

Now verify that modifications abide by insertion allowed "anywhere in existing production rules". We inserted terminal after 10 in rule_20. That is allowed. Additionally inserted terminal before 16 in rule_33. That's allowed.

Now ensure no other rule needed adjustment due to ordering constraints. Consider rule_20 after insertion: "A15 -> 10 23". Then A13 -> 8 A15 (raw). So A13 expands to "8 10 23". That's correct ordering.

Check if any other occurrence of A15 -> 10 23 may affect something else. When A13 expands to "8 A15", then A15 can also be "1" if we ever choose rule_19 (A15 -> 1). So those cases produce "8 1". That's okay; not used.

Now rule_33: A26 -> 22 16. So A26 can also use long production. When A26 used for A29 prefix we used rule_34, not rule_33. When used for A28 suffix we need "22 16". So choose rule_33. Works.

One more thing: In A28, we have A26 18. Since we modified rule_33 to "22 16", A28 yields 22 16 18. Perfect.

Now check if any other productions may cause ambiguous expansions that accidentally produce extra terminals not needed. But we only need existence of a derivation that yields target sequence; it's okay if there are alternative derivations.

Thus answer: list updated production rules: rule_20, rule_33.

Possibly also rule_20's original "A15 -> 10" changed to "A15 -> 10 23". Or we could also extend rule_20 to "A15 -> 10 23" exactly. Similarly rule_33: "A26 -> 22 16". That is all.

Thus final output.Updated production rules:
rule_20: A15 -> 10 23
rule_33: A26 -> 22 16