We need to consider a CFG with given productions. Terminal symbols are integers 1 to 36. The target sequence is a particular sequence of these integers that needs to be produced by the grammar starting from start symbol A1.

The grammar currently possibly cannot generate the target sequence because some terminal symbols have been removed from production rules. The only allowed operation is inserting terminal symbols anywhere in existing production rules (i.e., we can insert missing terminals into RHS of productions). We can't delete or reorder symbols, nor can we add new productions. We need to minimize total number of terminal symbols inserted.

Goal: adjust minimal number of RHS sequences, adding terminals (by placing them somewhere among existing symbols, respecting original order of existing symbols) so that there exists some derivation from A1 that yields exactly the target sequence.

We need to find minimal modifications.

We'll need to parse the grammar and target sequence, simulate possible derivations, identify missing terminals in each production that must appear to produce the target sequence.

Observation: The grammar appears to be designed to generate a particular sequence: The target sequence may be a concatenation of numbers that correspond to the terminals produced by various nonterminals.

We need to map each nonterminal's production to certain terminals to see which ones are missing. Perhaps it's a designed grammar that used to have a fully specified order, but some terminals were removed, resulting in missing pieces. We need to insert them back.

We can view each production, they may produce sequences, often consisting of a terminal followed by a nonterminal, like "36 A46 A49" etc. Probably each nonterminal corresponds to a segment of the target sequence.

Let's get the target:

TargetIndices:

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

We need to produce exactly this sequence.

Because we can't reorder nonterminals, we need to find the parse tree that yields this sequence, and missing terminals are those that were removed. Insert them in the appropriate productions.

Simplify: Because productions produce deterministic terminals (except choices), we can attempt to reconstruct parse tree from A1.

Production list:

rule_1: A1 -> A2
rule_2: A2 -> 36 A46 A49

Thus, the derivation begins with A1 => A2 => 36 A46 A49.

Thus the target must start with 36. Indeed the first terminal is 36. So far so good. After that we need to derive A46, then A49, producing the rest of the sequence: positions 2..end: "34 11 8 12 25 20 24 22 23 10 7 6 21 35 13 5 1 32 28 19 17 2 27 9 18 16 3 26 33 14 4 15 30 29 31". Let's denote rest as sequence S2.

We need to find production for A46, A49, and ensure their expansions produce S2 in order.

Now let's list rules for each nonterminal.

We have a large list. Let's map each nonterminal with its productions.

We'll go through all rules.

- A1 -> A2 (only rule)
- A2 -> 36 A46 A49 (only rule)
- A3 has two productions: 7 A4 and 8 A5 (rule_3, rule_4).
- A4 -> 1 | 2 | 3 | 6 (4 rules)
- A5 -> 1 | 2 | 3 (3 rules)
- A6 -> A7 A3 (rule_12)
- A7 -> 10 (rule_13)
- A8 -> A9 A10 (rule_14)
- A9 -> 11 (rule_15)
- A10 -> 7 A11 | 8 A12 (rule_16, rule_17).
- A11 -> 1 | 2 | 3 (rule_18-20)
- A12 -> 1 | 2 | 3 | 12 (rule_21-24)
- A13 -> A14 A15 (rule_25)
- A14 -> 13 (rule_26)
- A15 -> 4 A16 | 5 A17 (rule_27-28)
- A16 -> 1 | 2 | 3 (rule_29-31)
- A17 -> 1 | 2 | 3 (rule_32-34)
- A18 -> A19 A20 (rule_35)
- A19 -> 14 (rule_36)
- A20 -> 4 A21 | 5 A22 (rule_37-38)
- A21 -> 1 | 2 | 3 | 15 (rule_39-42)
- A22 -> 1 | 2 | 3 (rule_43-45)
- A23 -> A24 A25 (rule_46)
- A24 -> 18 (rule_47)
- A25 -> 16 A27 | 17 A26 (rule_48-49)
- A26 -> 1 | 2 | 3 (rule_50-52)
- A27 -> 1 | 2 | 3 (rule_53-55)
- A28 -> A29 A30 (rule_56)
- A29 -> 19 (rule_57)
- A30 -> 16 A31 | 17 A32 (rule_58-59)
- A31 -> 1 | 2 | 3 (rule_60-62)
- A32 -> 1 | 2 | 3 (rule_63-65)
- A33 -> 20 (rule_66)
- A34 -> 21 (rule_67)
- A35 -> 22 A36 (rule_68)
- A36 -> 9 | 23 (rule_69-70)
- A37 -> 25 A33 24 (rule_71)
- A38 -> 26 (rule_72)
- A39 -> 27 A40 (rule_73)
- A40 -> 9 (rule_74)
- A41 -> A42 A39 A23 A38 (rule_75)
- A42 -> 28 A28 (rule_76)
- A43 -> 29 (rule_77)
- A44 -> 30 A43 (rule_78)
- A45 -> 32 A41 (rule_79)
- A46 -> A47 (rule_80)
- A47 -> A48 A37 A35 A6 A34 (rule_81)
- A48 -> 34 A8 (rule_82)
- A49 -> A50 (rule_83)
- A50 -> A51 A45 A18 A44 (rule_84)
- A51 -> 35 A13 (rule_85)

We should try to see the hierarchical expansion.

Thus overall production chain:

A1 -> A2 -> 36 A46 A49

Now we need to derive A46 -> A47.

A47 -> A48 A37 A35 A6 A34 (in that order). So after 36 (which matches first terminal), we expand A47 using its production. So we need to derive a sequence that must start with whatever A48 yields, then A37, then A35, then A6, then A34.

Thus A48's RHS 34 A8: yields 34 and then whatever A8 expands to.

Thus after 36, next terminal should be 34 (coming from A48). Indeed second terminal in target is 34. Good.

So far the start matches: 36 (from A2), 34 (from A48).

Next we have A8 which is A9 A10. So after 34, we produce A9 then A10.

A9 yields 11. Indeed third terminal in target is 11. Good.

So A9 => 11, matches third.

Then A10 yields either 7 A11 (rule_16) OR 8 A12 (rule_17). We need to match the remainder of target starting at position 4: "8 12 25 20 24 22 23 10 7 6 21 35 13 5 1 32 28 19 17 2 27 9 18 16 3 26 33 14 4 15 30 29 31"

Let's examine possible expansions.

Option 1: A10 -> 7 A11 leads to terminal 7 then A11 produces whatever of {1,2,3}. Option 2: A10 -> 8 A12: yields terminal 8 then A12 which can produce {1,2,3,12}. Both could be possible.

Target at position 4 is 8, not 7. So we presumably need A10 -> 8 A12, with 8 matching position 4. Then A12 should produce some sequence; currently in the productions A12 -> 1 | 2 | 3 | 12, each produces a single terminal (no further nonterminals). So A12 can produce only one terminal. For A10 to produce more than one terminal (8 + something) we need further nonterminals. However, as per original grammar, A12 is a nonterminal that expands to a terminal only. So after 8 we will have a terminal, then after finishing A12 (producing a single terminal), we move on to the next part of the sequence, which starts after whatever A12 yields.

But target after 8 is "12". So we could have A12 -> 12. Indeed A12 has a production that yields 12. In the current grammar, A12 -> 12 is present (rule_24). So that matches target pos 5 = 12.

Thus A10 -> 8 A12, with A12 -> 12 yields "8 12". So far we have 36,34,11,8,12 matching target positions 1..5. Good.

After that we have processed up to terminal at position 5. Next we need to derive the rest: from position 6 onward: "25 20 24 22 23 10 7 6 21 35 13 5 1 32 28 19 17 2 27 9 18 16 3 26 33 14 4 15 30 29 31". Let's continue.

The remainder after A10 is still within A47's RHS: after A8, we have A37, A35, A6, A34.

So after we finish A8 (which gave us 13?? Let's compute A8 expansion: A8 -> A9 A10 => yields 11 then 8 12). So after A8 we have consumed up to position 5. Next in order is A37.

Thus A37 yields "25 A33 24". So A37 = 25 A33 24. Let's see what A33 and A33 produce: A33 -> 20. So A37 yields 25, then whatever A33 yields (20), then 24. So A37 yields "25 20 24". Indeed target positions 6-8 are "25 20 24". Perfect.

Thus A37 matches those three terminals.

Next after A37 we have A35. A35 -> 22 A36. So we get terminal 22 then A36.

A36 -> 9 | 23. So A35 yields either "22 9" or "22 23". Let's see target positions after 24: position 9 is 22, position 10 is 23. So we need A36 -> 23, yielding "22 23". Indeed we need A36 = 23. So choose that production. That matches positions 9 and 10.

Thus far we have matched positions 1 through 10.

Next after A35 is A6.

A6 -> A7 A3.

So after A35 (which gave 22 23) we go to A6 -> A7 A3.

A7 -> 10. So we output 10; position 11 is 10. Good.

Now A3 will produce either 7 A4 or 8 A5. Target next position after 10 (pos 12) is 7. So we likely need A3 -> 7 A4. So we output 7. Then A4 must produce something: A4 -> 1 | 2 | 3 | 6. The target after 7 is pos 13 = 6. So we need A4 -> 6. Indeed rule_8 gives A4 -> 6. That's fine. So A4 yields 6.

Thus A3 expansion yields "7 6". That matches target positions 12 and 13 (7 and 6). Good.

Now after A6 we have A34. A34 -> 21. So output 21 at next position (pos 14). Indeed pos 14 is 21. Good.

Thus after completing A34 we have consumed positions 1..14: 36 34 11 8 12 25 20 24 22 23 10 7 6 21.

Next the remaining part of the target (positions 15 onward): The rest is "35 13 5 1 32 28 19 17 2 27 9 18 16 3 26 33 14 4 15 30 29 31". Let's continue.

After A34, we have already processed all RHS of A47: A48 A37 A35 A6 A34. That's done. Next after A47 is the next part of the overall derivation: after A46 (which is A47), the next is A49. So we go to A49.

A49 -> A50.

Thus we go to A50.

Rule_84: A50 -> A51 A45 A18 A44

So A50 expands into A51, then A45, then A18, then A44.

Thus after previous matched segment we need to produce A51 (some terminals), then A45, then A18, then A44.

Goal: The next terminals (position 15 onward) should correspond to expansions of A51, etc.

Let's see: A51 -> 35 A13 (rule_85). So A51 outputs terminal 35 then expands A13.

Thus position 15 is 35. Indeed pos 15 = 35. Great.

Then after 35 we need to expand A13.

A13 -> A14 A15.

A14 -> 13.

Thus after 35, we get 13, then A15.

Thus positions 16 and onward: pos 16 is 13 (target 13). Good.

Now A15 -> either 4 A16 or 5 A17 (rules 27 and 28). Let's see target after 13: position 17 = 5. So we need A15 -> 5 A17.

Thus A15 yields terminal 5, then A17.

Thus after 13 we get 5. Position 17 = 5, good.

Now A17 -> 1 | 2 | 3 (rules 32-34). After 5, the next position is pos 18 = 1. So we need A17 -> 1.

Thus we get 1. Position 18 = 1, matches.

Thus A13 expansion yields "13 5 1". (Actually includes A14 =>13, then "5 1"). So A51 yields 35 13 5 1. So far we consumed positions 15-18 exactly.

Now we need to continue with A45 after A51.

A45 -> 32 A41 (rule_79). So A45 yields terminal 32, then expands A41.

Thus after position 18, we see position 19 is 32. Indeed pos 19 = 32. Good.

Thus we output 32 for A45 first terminal.

Now we need to expand A41.

A41 -> A42 A39 A23 A38 (rule_75). So after 32 we have A42 then A39 then A23 then A38 in order.

Thus we need to match the rest of the target sequence after 32 (starting at pos 20) using the expansions of those nonterminals in order.

Target positions after 32 (pos20 onward): 28, 19, 17, 2, 27, 9, 18, 16, 3, 26, 33, 14, 4, 15, 30, 29, 31.

Let's now enumerate the expansions.

First, A42 -> 28 A28 (rule_76). So A42 yields 28 then expands A28.

Thus position 20 should be 28. Indeed pos20 = 28.

Thus after 28 we need to expand A28.

A28 -> A29 A30 (rule_56). So we need to produce A29 then A30.

A29 -> 19 (rule_57). So terminal 19 next, matching pos21 = 19. Good.

Now A30 -> either 16 A31 or 17 A32 (rules 58, 59). After 19, next target pos22 = 17. So we need A30 -> 17 A32.

Thus we produce terminal 17 (pos22) and then expand A32.

A32 -> 1 | 2 | 3 (rules 63-65). Next target pos23 = 2. So A32 -> 2.

Thus after A30 we have consumed 17 2, matching positions 22-23.

Thus A28 yields "19 17 2". So far A42 yields "28 19 17 2". That's positions 20-23: 28 19 17 2. That matches target.

Now after A42 we go to A39.

A39 -> 27 A40 (rule_73). So A39 yields terminal 27, then expands A40.

Target pos24 = 27. Good.

A40 -> 9 (rule_74). So we get 9, target pos25 = 9. Good.

Thus A39 yields "27 9". Positions 24-25 match.

Now after A39 we have A23.

A23 -> A24 A25 (rule_46). So we need to expand A24 then A25.

A24 -> 18 (rule_47). So we produce terminal 18, target pos26 = 18. Yes.

Now A25 -> either 16 A27 or 17 A26 (rule_48,49). The next target pos27 = 16. So we need A25 -> 16 A27.

Thus we output terminal 16, then expand A27.

A27 -> 1 | 2 | 3 (rules 53-55). Next target pos28 = 3. So A27 -> 3.

Thus A25 yields "16 3". So A23 yields "18 16 3". Positions 26-28: 18 16 3. Indeed pos26 = 18, pos27 = 16, pos28 = 3 — they match the target.

But the explanation: Let's check target: At positions 26-28 we have:

pos26 = 18,
pos27 = 16,
pos28 = 3.

Hold on the earlier target listing had: after 27 9 we had "18 16 3". Indeed after 27 9 we have 18,16,3. So matches.

Now after A23 we have A38.

A38 -> 26 (rule_72). So we output terminal 26. target pos29 = 26. Yes.

Thus after A41's A42, A39, A23, A38 expansions, we have produced positions 20 through 29: 28,19,17,2,27,9,18,16,3,26. Indeed matches.

Thus after finishing A41, we have consumed up to pos29=26. At this point, we've completed A45 (which was A45 -> 32 A41). So we've consumed positions up to pos29 (including pos29). Actually we also need to consider A45 included 32 now before A41; we have consumed up to position 29 (which is terminal 26) after full expansion of A41.

Now after A45 we go back to A50's RHS (we have A51 already done, then A45, then A18, then A44). So next is A18.

A18 -> A19 A20 (rule_35). So we need to expand A19 then A20.

A19 -> 14 (rule_36). So we get terminal 14. Let's see target pos30 = 33. But target position after pos29 (26) is pos30 = 33. However we are expecting 14 from A19, not 33. That suggests something is off: perhaps we haven't accounted for A44 or something that yields 33,14,4, etc. Wait order: A50 = A51 A45 A18 A44. So after A45 we need to expand A18, and then A44. So after A45 we will produce some terminals for A18, then some terminals for A44.

What's target after position 29 (which consumed 26)? The target sequence continues:

Positions:
30: 33
31: 14
32: 4
33: 15
34: 30
35: 29
36: 31

Thus we have 7 terminals remaining: 33 14 4 15 30 29 31. The remaining nonterminals to expand are A18 then A44.

We need to see if A18 and A44 can produce these 7 terminals.

Let's analyze A18 and A44.

- A18 -> A19 A20. So A18 yields something from A19 then something from A20.

- A19 -> 14.

- A20 -> either 4 A21 or 5 A22.

Thus A18 can yield either "14 4 X" or "14 5 X", where X is from A21 or A22 respectively.

A21 -> 1 | 2 | 3 | 15

A22 -> 1 | 2 | 3

Thus A20 -> either "4 [1|2|3|15]" or "5 [1|2|3]".

Thus A18 can produce:

Case 1: 14 4 1 -> sequence: 14,4,1

Case 2: 14 4 2 -> 14 4 2

Case 3: 14 4 3 -> 14 4 3

Case 4: 14 4 15 -> 14 4 15

Case 5: 14 5 1 -> 14 5 1

Case 6: 14 5 2 -> 14 5 2

Case 7: 14 5 3 -> 14 5 3

Thus A18 yields exactly 3 terminals.

Now after A18, we have A44. A44 -> 30 A43 (rule_78). So it yields terminal 30 then expands A43.

A43 -> 29 (rule_77). So A44 yields "30 29". That's two terminals.

Thus combined A18 + A44 yields 5 terminals (if we consider only A18 produces 3, A44 produces 2). But we need to produce 7 remaining terminals: 33 14 4 15 30 29 31 (pos30-36). Actually we have over-run - we need also a leading 33 maybe from preceding something maybe missing from the grammar? Let's check if there is a nonterminal left that yields 33 and maybe something yields 31 at the end.

Looking at all nonterminals we haven't used: Well after A44 we may have something else? Actually A50's RHS ends at A44, which produces 30 29. Then after we finish A50, we have no further expansions. The entire derivation from A1 is:

A1 -> A2
A2 -> 36 A46 A49
A46 -> A47
A47 -> A48 A37 A35 A6 A34
A48 -> 34 A8
A8 -> A9 A10
...
A49 -> A50
A50 -> A51 A45 A18 A44

Thus final expansion includes A44 only as last bit. So after A44 we cannot generate any further terminals. However target has terminals "33 14 4 15 30 29 31". The last two 30 and 29 can be produced by A44 as we saw. The preceding ones may need to come from other productions that perhaps are missing some terminals that could be inserted. Also note "33" and "31" are not in the set produced by any right-hand side at the end. Let's search for nonterminals that could generate 33 and 31 anywhere else. There's no production that yields 33 (the integer 33) in any rule currently listed, except perhaps as a terminal that might be missing. Let's check list: We have productions that generate numbers: 1..36 appear in various rules:

From scanning:

- rule_1: none
...
- rule_2: 36
- rule_3: 7
- rule_4: 8
- rule_5: 1
- rule_6: 2
- rule_7: 3
- rule_8: 6
- rule_9: 1
- rule_10: 2
- rule_11: 3
- rule_12: none
- rule_13: 10
- rule_14: none
- rule_15: 11
- rule_16: 7
- rule_17: 8
- rule_18: 1
- rule_19: 2
- rule_20: 3
- rule_21: 1
- rule_22: 2
- rule_23: 3
- rule_24: 12
- rule_25: none
- rule_26: 13
- rule_27: 4
- rule_28: 5
- rule_29: 1
- rule_30: 2
- rule_31: 3
- rule_32: 1
- rule_33: 2
- rule_34: 3
- rule_35: none
- rule_36: 14
- rule_37: 4
- rule_38: 5
- rule_39: 1
- rule_40: 2
- rule_41: 3
- rule_42: 15
- rule_43: 1
- rule_44: 2
- rule_45: 3
- rule_46: none
- rule_47: 18
- rule_48: 16
- rule_49: 17
- rule_50: 1
- rule_51: 2
- rule_52: 3
- rule_53: 1
- rule_54: 2
- rule_55: 3
- rule_56: none
- rule_57: 19
- rule_58: 16
- rule_59: 17
- rule_60: 1
- rule_61: 2
- rule_62: 3
- rule_63: 1
- rule_64: 2
- rule_65: 3
- rule_66: 20
- rule_67: 21
- rule_68: 22
- rule_69: 9
- rule_70: 23
- rule_71: '25 A33 24' i.e., 25 and 24 are terminals, A33 = 20 yields 20 later.
- Actually rule_71: A37 -> 25 A33 24. So yields 25, <A33> (which is 20), and 24. So includes 25,20,24.
- rule_72: A38 26
- rule_73: A39 27 A40
- rule_74: A40 9
- rule_75: ... (maybe no terminals)
- rule_76: A42 28 A28, yields 28.
- rule_77: A43 29
- rule_78: A44 30 A43 produces 30 then whatever.
- rule_79: A45 32 A41 yields 32.
- rule_80: A46 A47 (none)
- rule_81: A47 A48 A37 A35 A6 A34 yield terminals 34 (via A48), and others in later expansions.
- rule_82: A48 34 A8 yields 34.
- rule_83: A49 A50
- rule_84: A50 A51 A45 A18 A44 yields no new terminals at start (A51 yields 35, A45 yields 32, A18 yields 14 etc.)
- rule_85: A51 35 A13 yields 35.

Thus we have explicit terminals from productions: 1-5? Actually 5 appears only as terminal in rule_28? (A15 -> 5 A17) Yes.

Check existence of terminal 33: I don't find a production that yields terminal 33. So 33 is missing. Possibly removed due to error. We need to insert 33 somewhere in an existing production rule.

Similarly, terminal 31 also not present. Let's search: terminal 31 appears in none of the productions. So 31 missing as well.

Thus 33 and 31 need to be inserted somewhere.

What about terminal 30 appears in rule_78 as part of A44. That's okay.

Terminal 31 appears only at end of target. So we need to add 31 to production somewhere after A44? Since after A44 we have no further productions. Maybe we could insert 31 into the RHS of A44 after its existing symbols: A44 -> 30 A43. A43 currently yields 29. So A44 expansions gives 30 29. And final terminal after that would be 30 29, not 31. We need to produce 31 after them. We could insert 31 after A43 in A44: i.e., A44 -> 30 A43 31 (or 30 A43 31, note insertion can be anywhere, not just at end). This would add a terminal 31 at the appropriate spot.

Similarly, where to insert 33? The target expects a 33 before 14. So "33 14 4 ...". So we need a 33 preceding the A18 (which yields 14 ...). The production A18 yields 14... at start of its expansion. So if we need 33 before that, we could insert 33 into the RHS of A50 before A18 (or maybe into A45 or earlier). A50's RHS is A51 A45 A18 A44. Currently expansions yield: from A51: 35 13 5 1, from A45: 32 ..., then A18: starting with 14..., then A44 yields 30 29 (plus maybe 31). So after A45 we have a 14. But we also need 33 to appear before that 14.

Thus best place: modify A50 to insert a 33 before A18. That is, A50 -> A51 A45 33 A18 A44. Insertion of terminal 33 between A45 and A18.

So that would cause after A45's expansion (ends with terminal from A41 final expansions), we produce 33, then A18's expansion (starting with 14). This would match target: after position 29 (terminal 26 we got from A38), we would have a new terminal 33 (pos30), then A18's 14 (pos31), then rest: probably "4 15" as part of A18's expansion (A20 -> ...) then A44's "30 29 31". Let's check.

So after insertion: A50 -> A51 A45 33 A18 A44.

Then expansion will be: (A51): 35 13 5 1, (A45): 32 <A41 expansion> which ends with terminal from A38 = 26 (position 29). Then we have inserted terminal 33 (pos30). Then A18 yields "14 ..." (pos31 etc). Then A44 yields "30 29 [maybe with 31]". Then final 31 inserted after A44 possibly.

We need to see the target after 33 is "14 4 15 30 29 31". Let's see if we can get "14 4 15 30 29 31" from A18 and A44 with minimal insertions (just 33 and 31 maybe). Let's compute.

A18 -> A19 A20. A19 yields 14 (pos31). Good.

A20 yields either "4 A21" or "5 A22". The target after 14 is "4 15". So we need A20 to be "4 A21" because 4 matches, and A21 needs to produce 15. Indeed A21 has a production 15 (rule_42). So A20 should be "4 A21" and A21 should be "15". So that gives "14 4 15". Good.

Thus after insertion of 33, using A18's rule 4 A21 with A21->15 yields "14 4 15". That's exactly positions 31-33 (14,4,15). Good.

Then after A18, we have A44 -> 30 A43. A43 -> 29. So that yields "30 29". Positions 34 and 35 should be 30 and 29, respectively. In target, positions 34 = 30, 35 = 29. Good.

Now we need terminal 31 at position 36. We can insert 31 after A44 using insertion into A44: e.g., modify A44 -> 30 A43 31 (insert 31 after A43). This yields "30 29 31". That matches positions 34-36.

Thus overall we need to insert terminal 33 into rule_84 A50 at appropriate spot. And insert terminal 31 into rule_78 A44.

Potentially we also need to consider if any other missing terminals exist. Let's check the full mapping of target sequence to derivation now with these insertions.

We'll re-evaluate entire derivation, confirming each position matches.

Define expansions now with modifications:

- rule_2 unchanged: A2 -> 36 A46 A49

- rule_80: A46 -> A47 (unchanged)

- rule_81: A47 -> A48 A37 A35 A6 A34 (unchanged)

- rule_82: A48 -> 34 A8 (unchanged)

- rule_14: A8 -> A9 A10 (unchanged)

- rule_15: A9 -> 11 (unchanged)

- rule_16: A10 -> 7 A11

- rule_17: A10 -> 8 A12 (choose this)

- rule_24: A12 -> 12 (unchanged)

- rule_71: A37 -> 25 A33 24 (unchanged). A33 -> 20 (rule_66)

- rule_68: A35 -> 22 A36 (unchanged)

- rule_70: A36 -> 23 (choose this)

- rule_13: A7 -> 10

- rule_3: A3 -> 7 A4 (choose this)

- rule_8: A4 -> 6 (we need 6 as terminal from A4)

- rule_67: A34 -> 21

- rule_84: A50 -> A51 A45 33 A18 A44 (modified from original A51 A45 A18 A44). Insert 33.

- rule_85: A51 -> 35 A13 (unchanged)

- rule_26: A14 -> 13

- rule_28: A15 -> 5 A17 (choose this)

- rule_34: A17 -> 1 (choose this)

- rule_79: A45 -> 32 A41

- rule_75: A41 -> A42 A39 A23 A38

- rule_76: A42 -> 28 A28

- rule_56: A28 -> A29 A30

- rule_57: A29 -> 19

- rule_59: A30 -> 17 A32 (choose)

- rule_64: A32 -> 2 (choose)

- rule_73: A39 -> 27 A40

- rule_74: A40 -> 9

- rule_46: A23 -> A24 A25

- rule_47: A24 -> 18

- rule_48: A25 -> 16 A27 (choose)

- rule_55: A27 -> 3

- rule_72: A38 -> 26

- rule_35: A18 -> A19 A20

- rule_36: A19 -> 14

- rule_37: A20 -> 4 A21 (choose) ; rule_42: A21 -> 15.

- rule_78: A44 -> 30 A43 31 (modified, inserted 31 after A43). Actually we have to modify rule_78 to insert 31. Since rule_78 originally is "A44 -> 30 A43". We'll change to "A44 -> 30 A43 31". This will emit terminals 30, 29 (from A43) and then 31.

- rule_77: A43 -> 29

Thus final sequence matches target.

Now we need to ensure no other missing terminals.

Let's parse the entire target with expansions to double-check all 36 positions.

Step-by-step:

1 A1->A2 => A2

A2: 36 A46 A49
- Output: 36 (pos1)
- Expand A46

A46 -> A47

A47 -> A48 A37 A35 A6 A34

Expand A48: "34 A8"
- Output: 34 (pos2)
- Expand A8

A8 -> A9 A10

A9: 11 (pos3)

A10 -> 8 A12 (pos4: 8)
A12 -> 12 (pos5: 12)

Thus far: positions 1-5: 36 34 11 8 12

Return to A47: Next expand A37.

A37 -> 25 A33 24

A33 -> 20

Output: 25 (pos6), 20 (pos7), 24 (pos8)

Now A35: "22 A36". Choose A36 -> 23 (pos9: 22, pos10: 23)

Now A6: A7 A3

A7 -> 10 (pos11)

A3 -> 7 A4 (pos12:7) and A4 -> 6 (pos13:6)

Now A34 -> 21 (pos14)

Now after A47's expansions, we've consumed pos1-14.

Now A49 -> A50

A50: original rule: A51 A45 A18 A44, but we've inserted 33 between A45 and A18.

Thus expansions:

A51 -> 35 A13

A13 -> A14 A15.

A14 -> 13 (pos16 = after 35? Wait ordering: A51 yields 35 then A13. So A51's output: pos15 = 35, pos16 = 13 (from A14). Wait we haven't output 35 yet. Let's start A50.

First A51:

- Output 35 (pos15)
- Expand A13:

A13 -> A14 A15

A14 -> 13 (pos16)
A15 -> 5 A17 (choose). Output 5 (pos17).
A17 -> 1 (choose). Output 1 (pos18).

Thus A51 yields 35,13,5,1. That matches pos15-18.

Then A45: 32 A41.

- Output 32 (pos19)
- Expand A41.

A41 -> A42 A39 A23 A38

Expand A42 -> 28 A28

- Output 28 (pos20)
- Expand A28 -> A29 A30

A29 -> 19 (pos21)
A30 -> 17 A32 (choose): output 17 (pos22)
A32 -> 2 (pos23)

Thus A42 yields 28 19 17 2 (positions 20-23)

Now next in A41: A39 -> 27 A40

- Output 27 (pos24)
- A40 -> 9 (pos25)

Now A41 next: A23 -> A24 A25

A24 -> 18 (pos26)
A25 -> 16 A27 (choose)

- Output 16 (pos27)
- A27 -> 3 (pos28)

Thus A23 yields 18 16 3 (pos26-28)

Now A41 next: A38 -> 26 (pos29)

Thus A41 yields positions 20-29 as we previously enumerated.

Thus A45's expansion yields 32 then all of A41's yields pos20-29.

Now after finishing A45, next in A50 is the inserted 33 (pos30). (modified)

Then A18: expands to A19 A20.

A19 -> 14 (pos31)

A20 -> 4 A21 (choose). Output 4 (pos32)

A21 -> 15 (pos33). (choose rule_42)

Thus A18 yields 14 4 15 (pos31-33)

Now finally A44: originally "30 A43", we inserted 31 after A43. So expand A44:

Output 30 (pos34)
A43 -> 29 (pos35)
Insert 31 (pos36)

Thus final positions 34-36 are 30 29 31.

Thus the entire target sequence matches with those two insertions.

Thus indeed we need to insert terminal 33 into rule 84 (A50). And insert terminal 31 into rule 78 (A44). No other missing terminals.

Now we need to confirm that there is no way to insert terminal 33 somewhere else that might reduce insertions (maybe combine with another rule?) But we still need to insert 31. Could we combine 33 insertion into an existing rule like A18 or A38? Let's see other possibilities.

The sequence after 26 is "33 14 4 15 30 29 31". One could modify A18 to produce 33 and 14 etc. But A18 currently yields A19 A20, which yields 14... So we could potentially insert 33 before the 14 inside A18's RHS: e.g., A18 -> 33 A19 A20. Since we can insert terminal symbols anywhere in existing production rules, we could modify rule_35 A18 -> A19 A20 to A18 -> 33 A19 A20, inserting 33 before A19. That would also produce 33 before 14. That would avoid inserting 33 into A50. However we must check if any other constraints. Inserting into A18 would require 33 inserted before A19's 14; it would produce "33 14 [rest]" consistent. However after A18 we have A44, which yields "30 29". Then we need to also insert 31 somewhere. Could we also insert 31 into A44 or somewhere else.

Thus we could consider (option 1) inserting 33 into rule_35: A18 -> 33 A19 A20, and keep A50 unchanged aside from maybe needed inserting 31 into A44. That would use two insertions as well (one for 33, one for 31). The originally we inserted 33 into rule_84. Both need two insertions total. Could both be inserted into a single rule? No, because 33 and 31 appear at different places (before 14 and after 29). There's no single production that can accommodate both.

Potentially we could modify A44 to produce "30 33 29 31"? No, 33 should be before 14, not after 30. So cannot combine.

Thus minimal number of insertions is at least 2 (one for 33, one for 31). Could we satisfy both with a single insertion? The two missing terminals are distinct and separated by other terminals in the target sequence: 33 appears at position 30 (before 14). 31 appears at the very end after 29. There is no rule that yields both positions, thus at least two insertions required.

Thus minimal number is 2.

Now we need to produce updated production rules that reflect these insertions, using the required format: "Updated production rules:" then list each rule that changed, with rule_<n>: and the new RHS.

Thus we need to modify rule_84 (A50) or rule_35 (A18) for 33, and rule_78 (A44) for 31.

The instruction: "Deletng or reordering symbols in existing production rules and creating new production rules is prohibited. Please try minimize the total number of terminal symbols inserted into the production rules" So insertion is only allowed within existing RHS.

Thus we can choose any rule for each insertion. We'll choose rule_84 for 33 and rule_78 for 31. Both involve modifications to a single rule each.

We need to be careful to preserve the shape: Insert 33 somewhere after A45 and before A18, but any position is okay as long as order of existing symbols remains same. Insert after A45 seems appropriate. Equivalent to "A50 -> A51 A45 33 A18 A44". (Add 33 between A45 and A18). We should not reorder existing symbols; the order A51 A45 A18 A44 stays, but now we have inserted a terminal (33) between A45 and A18. That's allowed.

Now for rule_78: "A44 -> 30 A43". We need to insert 31 somewhere. Options: after A43, or after 30, but consistent with target: we need 30, then 29, then 31. So we need to output A43 (29) before 31. So we should insert 31 after A43: "A44 -> 30 A43 31". That preserves order of existing symbols (30 then A43). We insert 31 after A43, allowed.

Thus we need to update two rules in output: rule_84 and rule_78.

But we must consider the numbering: The rules are named rule_1, rule_2... up to rule_85. rule_84 currently: "A50 -> A51 A45 A18 A44". So we need to update rule_84.

Similarly rule_78 currently: "A44 -> 30 A43". We'll update rule_78.

Now we need to ensure rule_84's RHS after insertion respects the original order, which it does: A51 A45 (original) then we insert terminal 33, then A18 A44.

Thus updated rule_84: "A50 -> A51 A45 33 A18 A44". For clarity we can put spaces as in original.

Similarly updated rule_78: "A44 -> 30 A43 31".

Now we need to check if any other missing terminals. Let's verify all terminals 1-36 appear in target and are generated somewhere by the grammar. The missing ones we identified are 33 and 31. Are there any other terminals missing from the grammar that appear in target? Let's verify each target terminal appears at least once initially:

List of target terminals: 36,34,11,8,12,25,20,24,22,23,10,7,6,21,35,13,5,1,32,28,19,17,2,27,9,18,16,3,26,33,14,4,15,30,29,31

The grammar original includes all except 33 and 31. Let's double-check each:

- 36: rule_2 provides 36.
- 34: rule_82 has 34.
- 11: rule_15.
- 8: rule_17 (8 A12) or rule_4.
- 12: rule_24.
- 25: rule_71 (A37 includes 25).
- 20: rule_66 yields 20 (via A33).
- 24: rule_71 includes 24.
- 22: rule_68 includes 22.
- 23: rule_70 includes 23.
- 10: rule_13 includes 10.
- 7: rule_3 includes 7, rule_16 includes 7.
- 6: rule_8 (A4 -> 6) indeed.
- 21: rule_67 yields 21.
- 35: rule_85 yields 35.
- 13: rule_26 yields 13.
- 5: rule_28 (A15 -> 5 A17) yields 5.
- 1: many rules produce 1 (A4->1 etc).
- 32: rule_79 yields 32.
- 28: rule_76 yields 28.
- 19: rule_57 yields 19.
- 17: rule_59 yields 17 as part of A30.
- 2: many produce 2 (A4->2 etc.)
- 27: rule_73 includes 27.
- 9: rule_69 gives 9 or rule_74.
- 18: rule_47 yields 18.
- 16: rule_48 yields 16.
- 3: many produce 3.
- 26: rule_72 yields 26.
- 33: missing
- 14: rule_36 yields 14.
- 4: rule_27 yields 4.
- 15: rule_42 yields 15.
- 30: rule_78 yields 30.
- 29: rule_77 yields 29.
- 31: missing

Thus only those two missing. So indeed minimal modifications are two insertions.

Now we need to write updated production rules in the given format: "rule_<n>: ..." for each changed rule.

Thus we output:

Updated production rules:
rule_84: A50 -> A51 A45 33 A18 A44
rule_78: A44 -> 30 A43 31

Make sure to include correct spaces and ordering as per original style.

Let’s double-check that we haven't altered rule numbers incorrectly. Accurate numbering: 

List original rules:

rule_78: A44 -> 30 A43 (as given)
rule_84: A50 -> A51 A45 A18 A44 (as given)

Thus modifications as above.

Now verify that inserting terminal 33 in rule_84 is allowed: Insert anywhere in existing production rules, preserving order of existing symbols. Inserting between A45 and A18 is allowed. So rule_84 becomes A51 A45 33 A18 A44. Good.

Inserting terminal 31 in rule_78: Insert after A43 is allowed: order 30 then A43 unchanged, and 31 inserted after. So new RHS: "30 A43 31". This retains original order for 30 A43. Good.

Thus final answer.

But we might want to confirm that the matrix of any other rule that could generate 33 or 31 appears to not produce them elsewhere, so you can't produce 33 and 31 using other rules without insertion. Indeed no rule has 33 or 31. So we need these insertions.

Thus final output. The instruction: "When you are finished please write a new line with `Updated production rules:`, and then write a new line for each of the updated production rules (including `rule_<n>:`), no need to rewrite the production rules which haven't changed. Do not write anything after the updated production rules."

Thus we need to output exactly:

Updated production rules:
rule_84: A50 -> A51 A45 33 A18 A44
rule_78: A44 -> 30 A43 31

We might want to preserve ordering of lines? Possibly list them in order of rule number (78 then 84). Could be fine either way. Since we have to list each updated rule including rule_<n>:. Let's list in ascending order: rule_78 then rule_84.

Thus final answer accordingly.Updated production rules:
rule_78: A44 -> 30 A43 31
rule_84: A50 -> A51 A45 33 A18 A44