We need to reconstruct the CFG by inserting terminal symbols into existing production rules (anywhere) to make the target sequence derivable from A1. The grammar includes many nonterminals and production rules. The terminals are integers 1-36. The target sequence includes many terminals, each integer appears at least once.

We have to insert terminals in existing productions but cannot reorder symbols, delete symbols, or create new rules. So we consider each existing rule's RHS, which may contain a mix of non-terminals and terminals, currently missing some necessary terminals.

Goal: find minimal insertions needed to accomplish derivation of given target sequence. The instructions: "Please try minimize the total number of terminal symbols inserted into the production rules so that the CFG can produce the target sequence." So we want minimal number of insertions.

We need to propose updated production rules with insertions.

Complex scenario: We have a large CFG, but likely the target sequence is exactly the concatenation of some of the productions in some order.

We need to fill missing terminals such that the sequence can be generated via the grammar rules, presumably the original grammar before some terminals were removed (the problem statement: "production rules have had some terminal symbols removed which may now render it impossible to produce the target sequence") and we must insert them back.

Thus we need to figure where each missing terminal goes.

We have a target sequence: Let's write it as list:

Index, terminal:
1:13
2:12
3:28
4:36
5:27
6:21
7:10
8:19
9:35
10:18
11:6
12:5
13:19
14:34
15:18
16:17
17:16
18:28
19:33
20:27
21:20
22:14
23:2
24:32
25:1
26:24
27:11
28:2
29:31
30:1
31:22
32:15
33:2
34:30
35:1
36:23
37:4
38:28
39:29
40:27
41:26
42:3
43:19
44:25
45:18
46:9
47:8
48:2
49:7
50:1

We need to derive this from start symbol A1.

Let's examine current grammar. Some productions produce terminals only: e.g., A4 -> 6 (but rule_6: A4 -> 6, meaning A4 yields terminal 6; missing maybe had other terminals? Similarly A5 -> 8 1, etc.)

One approach: Understand that each nonterminal currently expands to a fixed terminal string (maybe originally had a longer string). Since we cannot reorder symbols, we can insert terminals before or after any symbols in the RHS of any production.

We need to produce target sequence exactly, but grammar can produce many strings; we just need a derivation that yields the target. So we may want to map each part of target to expansions of A[x] in the order given by rule_1: A1 -> A43 A39 A36 A34 A30 A27 A23 A21 A10 A2

Thus A1 expands to a concatenation of expansions of those ten nonterminals: A43, A39, A36, A34, A30, A27, A23, A21, A10, A2.

Thus the target sequence must be formed as the concatenation of expansions of those in this order.

Thus we need each A_i to produce a substring of the target. Since we cannot change A1's RHS, we must assign substrings to each nonterminal accordingly.

Thus basically we need to parse the target string into 10 parts: the expansions of A43, A39, A36, A34, A30, A27, A23, A21, A10, A2.

The grammar includes multiple productions for each nonterminal, optional alternations.

Goal: Insert missing terminals into productions to make them exactly match the required substrings.

Thus we need to find a set of productions (perhaps picking one for each nonterminal) that can produce needed substrings, after inserting missing terminals.

We can adjust any existing productions by inserting terminals anywhere. Probably minimal insertions involve just adding missing terminals within the RHS of productions that currently yield some terminals.

Thus we need to decide for each nonterminal (e.g., A43) which production to use (rule_56 or rule_57) and adjust accordingly.

Let's enumerate productions for each relevant nonterminal:

- A43 -> A4 A44 (rule_56) or A7 A45 (rule_57). A4 currently yields 6 (terminal 6). A44 -> 5 28 27 (rule_58), and A7 -> 13 (rule_9). A45 -> 12 36 27 (rule_59).

Thus expansions:
* Using rule_56: A43 -> A4 A44 => 6 concatenated with 5 28 27 => "6 5 28 27"
* Using rule_57: A43 -> A7 A45 => 13 concatenated with 12 36 27 => "13 12 36 27"

But target starts with "13 12 28 36 27 ..." Wait target: 13,12,28,36,27... So near start we see "13 12 28 36 27". Options:

- A57 path yields "13 12 36 27": missing 28 after 12, and order of 36 and 27 after maybe? Actually yields 13 12 36 27, whereas target is 13 12 28 36 27.

Thus we need to insert terminal 28 between A1's A7 and A45 expansions? Wait A7 yields 13, A45 yields 12 36 27. So we need "13 12 28 36 27". That's the current string with missing 28 after 12 and maybe we have order "12 36 27". So we need to produce "12 28 36 27". So you could insert a 28 after the 12, and perhaps reorder doesn't allowed. But we can insert terminals within the RHS of rule A45: currently A45 -> 12 36 27. We can insert 28 between 12 and 36, resulting A45 -> 12 28 36 27. That yields exactly "12 28 36 27". Then A7->13 yields "13". So concatenation yields "13 12 28 36 27". Perfect. So we need to modify rule_59: A45 -> 12 28 36 27 (insert 28 after 12). Wait rule_59 currently: A45 -> 12 36 27. Insert 28 between 12 and 36. So A45 becomes 12 28 36 27. This adds one insertion.

Now check if target continues after "13 12 28 36 27". Next target terminal sequence: after 27 we have 21 10 19 35 18 etc. So after A43 (which covers first part), next is A39.

Let's compute A43's expansion now: Using rule_57 (A7 A45) yields "13 12 28 36 27". That matches target's first 5 terminals (13,12,28,36,27). So A43 part matches.

Thus A43 = rule_57, edge case: choose that production.

Now A39 expansion: A39 has productions: rule_50: A39 -> A13 A40 ; rule_51: A39 -> A18 A41 ; rule_52: A39 -> A20 A42

Let's examine each.

- A13 -> 21 (rule_18). So A13 yields 21.
- A40 -> 10 35 (rule_53). Yields 10,35.

Thus rule_50 leads to "21 10 35". That's target next three? After initial 5, target has: 21,10,19,35,18... Actually target after 27: 21,10,19,35,18,... So rule_50 only yields "21 10 35". Missing some terminals (like 19 after 10, and 18 after 35). Could we insert them? Yes, we can insert terminals within those productions. For example, insert 19 after 10 in A40? Actually A40 is "10 35". Insert 19 between 10 and 35 yields "10 19 35". Then sequence would be "21 10 19 35". Good. But still need "18" after 35, before next part. Next nonterminal after A39 is A36. So after A39 we need A36's expansion to produce 18 at the start perhaps.

Let's check target after "21 10 19 35". It continues: "18 6 5 19 34 18 17 16 ...". So likely A36 begins with 18 and maybe continues on.

Thus using A39 -> A13 A40 (rule_50) with inserted 19 after 10 in A40 yields "21 10 19 35". That matches the target's segment: after 27, we have 21 10 19 35. So that's good. No insertion needed for 21? That's already produced. Insert one terminal (19) into rule_53. That is insertion of terminal 19 after 10.

Alternatively, other options like A39 -> A18 A41 (rule_51). A18 yields 24 (rule_23). A41 yields 11 (rule_54). So "24 11". Not matching.

A39 -> A20 A42: A20 yields 26, A42 yields "3 19". So "26 3 19". Not matching.

Thus best is rule_50 with one insertion in rule_53.

But wait rule_53: A40 -> 10 35. Insert 19 between 10 and 35. That yields "10 19 35". Good.

Thus A39 yields "21 10 19 35". Confirm target: after 27 we have "... 21 10 19 35 18 ...". So A39 done.

Now next in A1: A36 (from rule_46 or 47). A36 -> A4 A37 (rule_46) or A9 A38 (rule_47). Let's see.

- A4 -> 6 (rule_6). So A4 yields 6.

- A37 -> 5 19 34 18 (rule_48). yields "5 19 34 18". So A36 via rule_46 yields "6 5 19 34 18". That's target after "21 10 19 35": next is "18 6 5 19 34 18". But we currently have sequence "6 5 19 34 18". There's no leading 18. So we need to produce "18 6 5 19 34 18". Could we produce "18 6 5 19 34 18"? Options: Insert a 18 before the 6 maybe, using rule_46? That would require inserting terminal 18 before A4. In rule_46, RHS is "A4 A37". We could insert 18 before A4: i.e., make RHS "18 A4 A37". That yields "18 6 5 19 34 18". That's exactly needed. So insertion of terminal 18 at start of rule_46.

Alternatively, using rule_47: A36 -> A9 A38. A9 -> 17 (rule_11). A38 -> 16 19 18 (rule_49). So yields "17 16 19 18". Not matching "18 6 ...". Could maybe insert many terminals, but seems more cost.

Thus we pick rule_46, insert 18 before A4, as described. So A36 yields "18 6 5 19 34 18". Good.

Thus far target: positions:

1-5: A43 (13 12 28 36 27)
6-9? No let's step.

Indices: Let's list again with segment breakdown.

Target: 
1:13
2:12
3:28
4:36
5:27
6:21
7:10
8:19
9:35
10:18
11:6
12:5
13:19
14:34
15:18
16:17
... Wait we consumed up to 15? Let's see after A36 we consumed terminals 10-15? Let's align:

Segment A43: terminals 1-5 = 13,12,28,36,27.

Segment A39: terminals 6-9 = 21,10,19,35.

Segment A36: we produced "18,6,5,19,34,18": that's terminals 10-15. Let's check: target[10]=18, [11]=6, [12]=5, [13]=19, [14]=34, [15]=18. Yes matches.

Thus after A36 we are at index 16: 17. Next nonterminal is A34 (from the list order A1: A43,A39,A36,A34...). Let's evaluate A34's productions.

Rule_44: A34 -> A9 A35. So only one production? Actually only rule_44 for A34; also A35 has rule_45.

Thus A34 expands to A9 followed by A35.

- A9 currently yields 17 (rule_11). So yields 17.

- A35 -> 16 33 27 (rule_45). That's "16 33 27". So A34 yields "17 16 33 27". That matches target after 15: At index 16 we need 17, 17:16, 18:28?, Actually let's check target after 15: positions from target:

List again with indices:

1:13
2:12
3:28
4:36
5:27
6:21
7:10
8:19
9:35
10:18
11:6
12:5
13:19
14:34
15:18
16:17
17:16
18:28
19:33
20:27
21:20
22:14
23:2
24:32
25:1
26:24
27:11
28:2
29:31
30:1
31:22
32:15
33:2
34:30
35:1
36:23
37:4
38:28
39:29
40:27
41:26
42:3
43:19
44:25
45:18
46:9
47:8
48:2
49:7
50:1

Thus after index 15 we have: 16:17, 17:16, 18:28, 19:33, 20:27, 21:20,...

Our A34 currently yields "17 16 33 27". That's matching indices 16 (17), 17 (16), then 18 (33) but target expects 28 at 18. So we have mismatch at position 18: we need 28 before 33. Also target after 33 is 27 (index 20) - that matches A34's last 27. So sequence we need: 17,16,28,33,27. Currently we have 17,16,33,27. So we need to insert terminal 28 between 16 and 33. That can be done in rule_45: A35 -> 16 33 27, by inserting 28 after 16 (or before 33). So modify rule_45 to "16 28 33 27". That adds one insertion.

Thus A34 yields "17 16 28 33 27". That matches indices 16-20: target 16:17,17:16,18:28,19:33,20:27. Good.

Thus we inserted one terminal.

Now next nonterminal in A1: A30.

A30 has three productions:
- rule_38: A30 -> A11 A31 
- rule_39: A30 -> A15 A32
- rule_40: A30 -> A16 A33

We need to match the target substring following index 20, which is at index 21 onward: "20 14 2 32 1 24 11 2 31 1 22 15 2 30 1 23 4 28 29 27 26 3 19 25 18 9 8 2 7 1".

Let's break them down:

Indices:
21:20
22:14
23:2
24:32
25:1
26:24
27:11
28:2
29:31
30:1
31:22
32:15
33:2
34:30
35:1
36:23
37:4
38:28
39:29
40:27
41:26
42:3
43:19
44:25
45:18
46:9
47:8
48:2
49:7
50:1

We need to see if A30 can be expanded to the start of this segment. Let's examine each production.

- A11 yields 20 (rule_16). So A11 -> 20.

- A31 yields "14 2 32 1" (rule_41). Right: rule_41: A31 -> 14 2 32 1. This matches indices 22-25 exactly: 14,2,32,1. So A30 via rule_38 (A11 A31) yields "20 14 2 32 1". That matches indices 21-25 (20 14 2 32 1). Great! So A30 can be rule_38 with no insertions.

Thus we choose rule_38.

Now next A1 nonterminal: A27.

A27 has productions: rule_34: A27 -> A16 A28 ; rule_35: A27 -> A18 A29.

We need to produce subsequent target substring after index 25, starting from index 26: "24 11 2 31 1 22 15 2 ...". Let's check each:

- A16 yields 23 (rule_21). So A16 -> 23.

- A28 yields 4 (rule_36). So A16 A28 yields "23 4". That's not matching start (24). So rule_34 not matching.

- A18 yields 24 (rule_23). So first token "24" matches index 26.

- A29 -> 11 2 31 1 (rule_37). Indeed rule_37: A29 -> 11 2 31 1. That's tokens: 11,2,31,1 which matches indices 27-30 (11,2,31,1). So A27 via rule_35 (A18 A29) yields "24 11 2 31 1". That matches indices 26-30. So good.

Thus no insertions needed.

Now next A1 nonterminal: A23.

A23 has productions: rule_28: A23 -> A13 A24 ; rule_29: A23 -> A15 A25 ; rule_30: A23 -> A16 A26.

We need to produce token sequence from index 31 onward: Starting at index 31: 22 15 2 30 1 23 4 28 29 27 26 3 19 25 18 9 8 2 7 1.

We need to see if any production fits.

Option 1: Using rule_28: A13 A24. A13 -> 21 (but we need 22). So not matching.

Option 2: rule_29: A15 A25. A15 -> 22 (rule_20). Good! A15 yields 22, which matches index 31 (22). Then A25 -> 15 2 30 1 (rule_32). That yields "15 2 30 1". That's exactly indices 32-35: 15,2,30,1. So rule_29 yields "22 15 2 30 1". That matches indices 31-35. Perfect.

Thus A23 can be rule_29 with no insertions.

Option 3: rule_30 (A16 A26): A16->23 gives "23" but we need "22" first, so not needed.

Thus we choose rule_29.

Now next A1 nonterminal: A21.

Production: rule_26: A21 -> A16 A22.

A16 -> 23 (as before). A22 -> 4 (rule_27). So yields "23 4". Index after A23 we consumed up to index 35. Next target indices: 36:23, 37:4. Indeed we have 23, 4. So A21 yields exactly "23 4". Good match. No insertions.

Now next A1 nonterminal: A10.

A10 has productions: rule_12: A10 -> A13 A12 ; rule_13: A10 -> A15 A14 ; rule_14: A10 -> A18 A17 ; rule_15: A10 -> A20 A19

We need to produce substring from index 38 onward: target indices 38 to 50 = "28 29 27 26 3 19 25 18 9 8 2 7 1".

Let's see each production.

But we need to map expected strings.

We have A13 -> 21 ; A12 -> 10 19 18 (rule_17). So rule_12 yields "21 10 19 18". That doesn't match.

A15 -> 22 ; A14 -> 15 19 18 (rule_19). So rule_13 yields "22 15 19 18". Not matching.

A18 ->24 ; A17 -> 11 19 18 (rule_22). yields "24 11 19 18". Not matching.

A20 -> 26 ; A19 -> 3 19 25 18 (rule_24). yields "26 3 19 25 18". That matches part of target: indices 41-45: "26 3 19 25 18" appear at indices 41:26, 42:3, 43:19, 44:25, 45:18. But we need a prefix before that: indices 38-40: 28 29 27. So perhaps A10 uses rule_15 (A20 A19) but we need to add "28 29 27" before the A20? Actually A20 leads to 26, not 28. So maybe we need to combine a production that starts with "28 29 27"? Let's check other productions.

But perhaps there are also alternative expansions for A10 but we can't create new. However we can insert terminals into existing RHS anywhere. So we could choose for A10 rule_15 (A20 A19) and insert terminals 28 29 27 before A20 (or within A20). The RHS of rule_15 is "A20 A19". We could add terminals before A20 and/or between A20 and A19, as well as inside A20's RHS. Since A20 currently yields "26". We need to produce "28 29 27 26 3 19 25 18". That's exactly 8 terminals. Our current rule_15 yields "26 3 19 25 18". To get prefix 28 29 27, we could insert them before A20, but that would be after A10's rule? Actually we can modify rule_15 to "28 29 27 A20 A19". Insert three terminals 28, 29, 27 before A20. Then we get "28 29 27 26 3 19 25 18". That's exactly target indices 38-45. Then after A10 we have next nonterminal in A1: A2. So this would match up to index 45.

Now we must check target after index 45: after 18 (index45), we have "9 8 2 7 1". That's the final segment. So A2 must produce that. Let's examine A2.

A2 has productions: rule_2: A2 -> A4 A3 ; rule_3: A2 -> A6 A5 ; rule_4: A2 -> A9 A8

We need to produce "9 8 2 7 1". Let's see expansions:

- A4 -> 6, A3 -> 5 2 (rule_5). That yields "6 5 2". Not matching.

- A6 -> 9, A5 -> 8 1 (rule_7). That yields "9 8 1". That's close: we need "9 8 2 7 1". Actually the A6 A5 combo yields 9 8 1 (three terminals). We need extra "2 7". Could we insert those? Perhaps insert "2 7" between A6 and A5, or after A5 before something? But rule_3's RHS is "A6 A5". We could insert "2 7" between A6 and A5, resulting "9 2 7 8 1"? But order would be A6 ->9, then new terminals 2 7, then A5 -> 8 1 yields "9 2 7 8 1". That's not matching sequence, because target expects 9,8,2,7,1. So we need "9 8 2 7 1". That is A6 (9), then A5 yields "8 1", but we need 8, then 2,7, then 1. So we need to insert "2 7" between the 8 and 1, i.e., inside A5's RHS (which currently is "8 1"). We could modify rule_7: A5 -> 8 1, insert "2 7" after 8 before 1: yields "8 2 7 1". Then A6 A5 yields "9 8 2 7 1". That's exactly target. So we need to insert terminals 2 and 7 into rule_7.

Thus using rule_3 (A2 -> A6 A5) and modify rule_7 to be "8 2 7 1". That adds two insertions.

Check alternative A2 productions:

rule_4: A2 -> A9 A8: A9 yields 17, A8 yields 16 2 (rule_10). That's not matching.

Thus best is rule_3 with inserted terminals.

Thus overall modifications:

- Insert 28 after 12 in rule_59 (A45 ->12 28 36 27) [1 insertion]
- Insert 19 after 10 in rule_53 (A40 ->10 19 35) [1 insertion]
- Insert 18 before A4 in rule_46 (A36 ->18 A4 A37) [1 insertion]
- Insert 28 after 16 in rule_45 (A35 ->16 28 33 27) [1 insertion]
- Insert 28, 29, 27 before A20 in rule_15 (A10 -> 28 29 27 A20 A19) [3 insertions]
- Insert 2, 7 after 8 in rule_7 (A5 -> 8 2 7 1) [2 insertions]

Total insertions: 1+1+1+1+3+2 = 9 insertions.

Check all connections and ensure no conflict.

Let's verify full derivation step by step:

Start: A1

A1 -> A43 A39 A36 A34 A30 A27 A23 A21 A10 A2

Now expansions:

- A43 -> A7 A45 (choose rule_57)
  - A7 -> 13
  - A45 -> 12 28 36 27 (modified)
Thus yields 13 12 28 36 27.

- A39 -> A13 A40 (rule_50)
  - A13 -> 21
  - A40 -> 10 19 35 (modified)
Thus yields 21 10 19 35.

- A36 -> 18 A4 A37 (modified rule_46)
  - 18 (inserted)
  - A4 -> 6
  - A37 -> 5 19 34 18
Thus yields 18 6 5 19 34 18.

- A34 -> A9 A35 (rule_44)
  - A9 -> 17
  - A35 -> 16 28 33 27 (modified)
Thus yields 17 16 28 33 27.

- A30 -> A11 A31 (rule_38)
  - A11 -> 20
  - A31 -> 14 2 32 1
Thus yields 20 14 2 32 1.

- A27 -> A18 A29 (rule_35)
  - A18 -> 24
  - A29 -> 11 2 31 1
Thus yields 24 11 2 31 1.

- A23 -> A15 A25 (rule_29)
  - A15 -> 22
  - A25 -> 15 2 30 1
Thus yields 22 15 2 30 1.

- A21 -> A16 A22 (rule_26)
  - A16 -> 23
  - A22 -> 4
Thus yields 23 4.

- A10 -> 28 29 27 A20 A19 (modified rule_15)
  - Inserted terminals: 28 29 27
  - A20 -> 26
  - A19 -> 3 19 25 18
Thus yields 28 29 27 26 3 19 25 18.

- A2 -> A6 A5 (rule_3)
  - A6 -> 9
  - A5 -> 8 2 7 1 (modified)
Thus yields 9 8 2 7 1.

Now concatenating everything:

A43: 13 12 28 36 27
A39: 21 10 19 35
A36: 18 6 5 19 34 18
A34: 17 16 28 33 27
A30: 20 14 2 32 1
A27: 24 11 2 31 1
A23: 22 15 2 30 1
A21: 23 4
A10: 28 29 27 26 3 19 25 18
A2: 9 8 2 7 1

Let's list them in order:

1-5:13 12 28 36 27
6-9:21 10 19 35
10-15:18 6 5 19 34 18
16-20:17 16 28 33 27
21-25:20 14 2 32 1
26-30:24 11 2 31 1
31-35:22 15 2 30 1
36-37:23 4
38-45:28 29 27 26 3 19 25 18
46-50:9 8 2 7 1

Now write out entire sequence with indices to compare to target:

Indices:

1:13
2:12
3:28
4:36
5:27
6:21
7:10
8:19
9:35
10:18
11:6
12:5
13:19
14:34
15:18
16:17
17:16
18:28
19:33
20:27
21:20
22:14
23:2
24:32
25:1
26:24
27:11
28:2
29:31
30:1
31:22
32:15
33:2
34:30
35:1
36:23
37:4
38:28
39:29
40:27
41:26
42:3
43:19
44:25
45:18
46:9
47:8
48:2
49:7
50:1

Now compare to target earlier:

Target:

1:13
2:12
3:28
4:36
5:27
6:21
7:10
8:19
9:35
10:18
11:6
12:5
13:19
14:34
15:18
16:17
17:16
18:28
19:33
20:27
21:20
22:14
23:2
24:32
25:1
26:24
27:11
28:2
29:31
30:1
31:22
32:15
33:2
34:30
35:1
36:23
37:4
38:28
39:29
40:27
41:26
42:3
43:19
44:25
45:18
46:9
47:8
48:2
49:7
50:1

Matches exactly! Great.

Thus a solution with 9 insertions works. We need to output the updated production rules where modifications occurred.

List all the rules we changed:

rule_59: was "A45 -> 12 36 27". Change to "A45 -> 12 28 36 27". Insert 28.

rule_53: "A40 -> 10 35". Change to "A40 -> 10 19 35". Insert 19.

rule_46: "A36 -> A4 A37". Change to "A36 -> 18 A4 A37". Insert 18 before A4.

rule_45: "A35 -> 16 33 27". Change to "A35 -> 16 28 33 27". Insert 28.

rule_15: "A10 -> A20 A19". Change to "A10 -> 28 29 27 A20 A19". Insert 28 29 27 before A20.

rule_7: "A5 -> 8 1". Change to "A5 -> 8 2 7 1". Insert 2 7 after 8 (and before 1). Actually "8 2 7 1".

Now check any other modifications needed: Did we need to modify any other rules? Let's check rule_5 - maybe not needed. So that's all.

We'll output "Updated production rules:" then each changed rule with label rule_<n>: ... as required.

But ensure format matches exactly: "rule_59: A45 -> 12 28 36 27". Or perhaps need to keep spaces.

Should we write each updated rule with insertion? Yes.

Thus final answer will be:

Updated production rules:
rule_59: A45 -> 12 28 36 27
rule_53: A40 -> 10 19 35
rule_46: A36 -> 18 A4 A37
rule_45: A35 -> 16 28 33 27
rule_15: A10 -> 28 29 27 A20 A19
rule_7: A5 -> 8 2 7 1

We must not write anything after the updated production rules.

One nuance: rule_7 originally: A5 -> 8 1 . Inserted 2, 7. In order we insert them: "8 2 7 1". It seems correct.

Double-check that all the inserted terminals are valid (1-36). 28, 29, 27, 18, 16, 2, 7, 19 all are within 1-36. Yes.

Thus output.

Before delivering final answer ensure that we didn't overlook any needed modifications.

Check all productions used: A45, A40, A36, A35, A10, A5. Yes.

Also note that rule_59 hasn't been used in any other production aside from A45, which is only used in A43 -> A7 A45. Good.

Now must ensure that there are no multiple productions for A45 other than rule_59; that's fine.

Now check that the inserted terminals didn't affect any other strings. Since we only care about this derivation, it's okay.

One minor check: In rule_46 we inserted 18 before A4. The rule RHS only contains nonterminals A4 A37; we inserted 18 before A4, so the order is "18 A4 A37". That's allowed (anywhere). Ensure there is no other rule that uses A36 -> A4 A37 differently; there are only two productions: rule_46 and rule_47. Our chosen rule_46 is modified.

Thus overall modifications are minimal.

Now produce final answer.Updated production rules:
rule_59: A45 -> 12 28 36 27
rule_53: A40 -> 10 19 35
rule_46: A36 -> 18 A4 A37
rule_45: A35 -> 16 28 33 27
rule_15: A10 -> 28 29 27 A20 A19
rule_7: A5 -> 8 2 7 1