We have a CFG where terminal symbols 1..36 (integers). The grammar originally had some terminals, but some might have been removed (maybe due to a scenario where each rule originally had some sequence of terminals, but some have been removed). We need to insert missing terminals into existing production rules, without deleting any, reorder, or creating new rules. The objective is to make the grammar able to generate the target sequence exactly. We need to minimize the total number of inserted terminals.

We are given a CFG with 63 rules. Many are only nonterminals (e.g., A1 -> …). Some nonterminals have multiple alternative productions (like A2 -> A4 A3 and A2 -> A8 A7). We'll need to find a parse tree deriving the target sequence. However, some parts of the target sequence may be missing in the grammar (maybe terminals have been removed from productions). We need to fix it by adding missing terminal symbols into productions (anywhere in RHS). But cannot delete or rearrange existing symbols. Cannot add new production rules. We have to use the existing structure.

Goal: minimize inserted terminals (they count across all rules). Need to produce exactly the target sequence "11 10 2 36 1 25 12 2 35 1 26 7 2 34 1 9 8 32 33 31 17 16 21 30 20 19 18 21 29 20 24 13 2 28 1 15 14 21 27 20 23 3 21 22 20 6 5 2 4 1".

Let's list the target sequence:

Indices:

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

We have 50 symbols. We need to produce that from start symbol A1.

We need to see the grammar's structure: It seems like a binary tree arrangement, where each nonterminal expands to other nonterminals or sequences of terminals (numbers). Typically each rule maps either nonterminal -> sequence of terminals (like rule_4: A3 -> 5 2) or a mix of terminals and nonterminals (like rule_5: A4 -> 6). Actually rule_5 is A4 -> 6 (so it's a 1-terminal rule). Many rules produce sequences of terminals (sometimes 1, sometimes more). Some are of the form A* -> A* A*, where both are nonterminals. Others are A* -> terminal terminal ... extended.

Goal: find a derivation of the target from A1.

Currently, does any derivation produce something? Probably the original grammar had all required terminals, and this target was originally derivable. Since we now have missing terminals removed, we need to re-add them. For each rule we can insert additional terminals anywhere in RHS. That can help correct expansions.

We have to minimize total insertions. So we should try to reuse existing terminals as much as possible; only add missing ones.

Thus we need to perform a parsing of the target, matching known RHS sequences, and wherever they diverge we insert necessary terminals.

We'll need to decide which alternative productions to use for each nonterminal.

Let's outline the grammar.

Start symbol A1 -> A43 A40 A37 A33 A29 A24 A20 A18 A11 A2

That's a chain of 10 nonterminals in that exact order. So the target sequence must be concatenation of the yields of A43, then A40, then A37, then A33, then A29, then A24, then A20, then A18, then A11, then A2 (in that order).

Thus the target should be partitioned into 10 substrings, each derived from those nonterminals.

Thus we need to find a segmentation of target sequence of length 50 into 10 pieces that match these nonterminals.

Let's denote the target as T[1..50]. We'll try dividing:

- Subseq1: A43 yields prefix of T.
- Next A40 yields next chunk.
- Next A37 yields next.
- Next A33 yields next.
- Next A29 yields next.
- Next A24 yields next.
- Next A20 yields next.
- Next A18 yields next.
- Next A11 yields next.
- Finally A2 yields suffix.

We need to find yields of each nonterminal via the production rules.

Given the target, we can try to map known productions that produce some terminal sequences.

List known productions:

Let’s put each nonterminal and possible productions.

A1: given above.

A2: two alternatives:
- rule_2: A2 -> A4 A3
- rule_3: A2 -> A8 A7

Thus A2 yields concatenation of yields of A4 and A3 (or A8 then A7).

A3 -> 5 2 (rule_4). So A3 yields 5 2.

A4 -> 6 (rule_5). So A4 yields 6.

Thus using rule_2, A2 yields 6 5 2 (i.e., 6 5 2). That's a known sequence.

A8 -> 15 (rule_9). So A8 yields 15.

A7 -> 14 2 (rule_8). So A7 yields 14 2.

Thus rule_3 (A2 -> A8 A7) yields 15 14 2. That is 15, 14, 2.

Thus A2 can produce either [6,5,2] or [15,14,2].

Check the tail of target: the last three numbers are "4 1"? Actually last 3 are positions 48,49,50: 2,4,1. The last three numbers are 2 4 1. The second to last is 4 (terminal 4). The target ends with "... 5 2 4 1". Actually let's confirm: position 46:6, 47:5, 48:2, 49:4, 50:1.

Thus tail of target is "...6 5 2 4 1". So it's like "6 5 2 4 1". A2 could produce "6 5 2". So that matches the "6 5 2" part near the end. After that, we have "4 1". So maybe A2 is used to match "6 5 2" leaving remainder "4 1" which might be considered part of A?? However maybe A2 -> A4 A3 yields exactly "6 5 2". That's indeed positions 46-48: 6,5,2. Then after A2, there would be something else? Actually A2 is the final nonterminal in A1's expansion, so after A2 there is no more stuff. Thus the overall sequence derived from A1 will include whatever A2 yields as final part. According to target, the final part after A2 (i.e., from A2) would need to match the suffix "... 6 5 2 4 1". But A2 yields only "6 5 2" (or "15 14 2"). So we cannot get "4 1" after that. However we might be able to insert missing terminals into the production rules to allow A2 to yield "6 5 2 4 1". For instance, we could insert "4 1" into the production for A2 (somewhere inside either rule_2 or rule_3). Which is allowed: we can insert terminals anywhere in RHS; thus we could change rule_2 to be, e.g., A2 -> A4 4 1 A3 or A4 A3 4 1 or any placement. But we must minimize total inserted terminals.

Another possibility: A2's production already might produce "6 5 2 4 1" if we have extra inserted terminals into A4 -> 6 (maybe add "6 5"? No, re-evaluate). Actually A4 yields 6, but we could add extra terminals before/after "6" in A4's RHS? Wait A4 is a nonterminal, rule_5: A4 -> 6. We can insert terminals into RHS of rule_5, e.g., A4 -> 6 4 1. That would make A4 produce 6 4 1. Then A2 -> A4 A3 yields "6 4 1 5 2". That would be 6 4 1 5 2, which does not match "6 5 2 4 1". We could also insert into A3: currently A3 -> 5 2, could become 5 2 4 1, then A2 (A4 A3) yields 6 5 2 4 1, which matches suffix "6 5 2 4 1". Indeed the suffix is 6,5,2,4,1. So insertion in A3 could fix it. Insert "4 1" after the 5 2? Actually after 5 2 we need "4 1". So we could modify rule_4 (A3 -> 5 2) to A3 -> 5 2 4 1. That satisfies suffix exactly.

Alternatively we could use rule_3 (A2 -> A8 A7) to produce something else; but that gives 15 14 2 maybe not match.

Thus likely we will use A2 -> A4 A3, and modify rule_4 to have extra 4 1, which adds 2 insertions.

We need to minimize total inserts. So far 2 inserts.

Now we need to parse the rest for other nonterminals.

Let's now find other productions.

A5 -> 9 (rule_6). So yields 9.
A6 -> 11 (rule_7). So yields 11.
A7 -> 14 2 (rule_8). Yields 14 2.
A8 -> 15 (rule_9). yields 15.
A9 -> 17 (rule_10). yields 17.
A10 -> 19 (rule_11). yields 19.
A11 -> A13 A12 (rule_12) or A16 A15 (rule_13) (two alternatives).
A12 -> 3 22 20 (rule_14) yields 3 22 20.
A13 -> 23 (rule_15) yields 23.
A14 -> 24 (rule_16) yields 24.
A15 -> 12 21 20 (rule_17) yields 12 21 20.
A16 -> 25 (rule_18) yields 25.
A17 -> 26 (rule_19) yields 26.
A18 -> A8 A19 (rule_20). So yields A8 then A19.
A19 -> 14 21 27 (rule_21). yields 14 21 27.
A20 -> options: A13 A21 (rule_22), A14 A22 (rule_23), A16 A23 (rule_24).
- A21 -> 3 1 (rule_25).
- A22 -> 13 2 28 (rule_26).
- A23 -> 12 2 1 (rule_27).
A24 -> options: A5 A25 (rule_28), A8 A26 (rule_29), A9 A27 (rule_30), A10 A28 (rule_31).
- A25 -> 8 (rule_32).
- A26 -> 14 20 (rule_33).
- A27 -> 16 (rule_34).
- A28 -> 18 21 29 20 (rule_35).
A29 -> options: A8 A30 (rule_36), A9 A31 (rule_37), A10 A32 (rule_38).
- A30 -> 14 21 20 (rule_39).
- A31 -> 16 30 (rule_40).
- A32 -> 18 21 20 (rule_41).
A33 -> options: A5 A34 (rule_42), A6 A35 (rule_43), A8 A36 (rule_44).
- A34 -> 8 33 31 (rule_45).
- A35 -> 10 32 31 (rule_46).
- A36 -> 14 (rule_47).
A34 yields terminals: 8 33 31 (so terminal 33 is a terminal! Note: we have terminal symbol 33; that's required in target sequence, appears at position 19: 33. Good). Actually this rule is A34 -> 8 33 31. That yields 3 terminals.

A35 yields 10 32 31. Terminals 10, 32, 31. The target contains "10 2 ..."? Actually target has many 10's and 31's. Check: position 2 is 10, later at pos 20 we have 31, at pos 18 we have 32, also at pos 31 we have 24, etc.

A36 yields 14 (one terminal). That's in target at position 36: 14.

A37 -> alternatives: A16 A38 (rule_48) or A17 A39 (rule_49).
- A38 -> 12 2 1 (rule_50). So yields 12 2 1.
- A39 -> 7 2 34 1 (rule_51). yields 7 2 34 1.

A38 yields 12 2 1: matches positions around 6-8? The target has 25 at position 6, then "12 2 ...". In target we have "25 12 2 ..." after position 6: Actually target positions: 1 11, 2 10, 3 2, 4 36, 5 1, 6 25, 7 12, 8 2, 9 35, 10 1, ... So we need to produce sequence of "25 12 2 35 1". That may be from A37? Let's examine: A37 -> A16 A38 is 25 then 12 2 1 = yields 25 12 2 1. That's missing 35 in between. The target has 25 12 2 35 1. So we need to get 35 after 2 and before 1. We could insert 35 somewhere: maybe into A38's production. A38 currently yields 12 2 1; we could insert 35 between 2 and 1. So A38 -> 12 2 35 1 (inserting terminal 35). That's one insertion (35). Then A16 = 25, then A38 yields 12 2 35 1 => overall 25 12 2 35 1. That matches the target segment positions 6 to 10: 25 12 2 35 1.

Thus we need to insert "35" into rule_50 (A38). So +1 insertion.

Alternatively could use other production A37 -> A17 A39, which yields 26 then 7 2 34 1 = 26 7 2 34 1. That matches positions 11-15: 26 7 2 34 1. Indeed target at positions 11-15 is exactly 26, 7,2,34,1. So we will need both A37 variations? Actually A37 appears just once as part of A1 expansion (A37 after A40). But A37 may have a single choice for its derivation. Because there are two alternatives for A37; we must pick one. But target comprises both "25 12 2 35 1" and later "26 7 2 34 1". Wait, those are two distinct substrings: 25 12 2 35 1 (positions 6-10) and then 26 7 2 34 1 (positions 11-15). These correspond to A37? Actually A37 appears once in A1's RHS. So A37 must produce a continuous substring that contains both 25 and 26 parts. But currently it cannot produce both; it can produce either 25 12 2 35 1 (if using A16 A38) or 26 7 2 34 1 (if using A17 A39). So we cannot produce both unless we modify grammar to embed extra terminal (like maybe A37 -> A16 A38 ... but also some other path). However we have only one A37, and we need to produce A37's yield matching the target segment from after A40 before A33. Let's locate positions accordingly.

Sequence of nonterminals: A1 -> A43, A40, A37, A33, A29, A24, A20, A18, A11, A2.

Thus target order must be:

- segment from A43: some prefix,
- then A40,
- then A37,
- then A33,
- then A29,
- then A24,
- then A20,
- then A18,
- then A11,
- then A2.

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

Let's see the plausible partition. On quick glance, known sequences:

- A43 expansions: It can be A5 A44, A6 A45, A8 A46, A10 A47.

A5 -> 9. So A5 A44 yields 9 + whatever A44 yields.

A44: rule_60: A44 -> 8 2 1.

Thus A5 A44 yields 9 8 2 1. That matches a substring "9 8 2 1"? In target we do see "9 8 32 33 31...", but "9 8" is at position 16-17: 9 (pos16) 8 (pos17). But after those we have 32 33 31, not "2 1". So maybe not that.

A6 -> 11. So A6 A45 yields 11 + A45.

A45 rule_61: A45 -> 10 2 1.

Thus yields 11 10 2 1. In target we start with "11 10 2 36 1". That's close: we have 11 10 2, then 36 and then 1. So A6 A45 gives 11 10 2 1, missing "36" before the final 1. Also after that, we have "36" at position 4, and "1" at position 5. So if we could insert "36" inside A45 or after something. A45 yields "10 2 1". Maybe we can insert "36" before the final 1 in A45 (or after "2")? A45's RHS is "10 2 1". We could insert "36" between "2" and "1". So A45 becomes "10 2 36 1". That adds a single insertion (36). Resulting A6 A45 yields "11 10 2 36 1". That's exact prefix of target positions 1-5.

Thus A43 could be via production "A6 A45". Then A43 yields "11 10 2 36 1". Good.

Thus we choose A43 = A6 A45 using rule_57 (A43 -> A6 A45). This matches prefix of target.

Thus we need to insert "36" into rule_61 (A45) after "2". So rule_61 is A45 -> 10 2 1, we insert "36" before the 1. Revised: A45 -> 10 2 36 1.

Now we have prefix solved.

Now next nonterminal is A40. Need to produce substring after that, which is presumably positions 6 onward: positions 6-? Let’s list after prefix: we have consumed positions 1-5: 11, 10,2,36,1.

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

Positions 6..50.

So A40 must generate a prefix of the remaining.

A40 has options: A14 A41 (rule_52) or A16 A42 (rule_53). Let's examine each.

- A14 -> 24 (rule_16). So A14 yields 24.

- A41 -> 13 2 1 (rule_54). Yields 13 2 1.

Thus A14 A41 yields 24 13 2 1. In target we see "24 13 2 28 1 ..." Actually at position 31 is 24, then positions 32-34 are 13 2 28, then 35 is 1. So A14 A41 yields 24 13 2 1. Need "28" inserted between "2" and "1". Could be inserted in A41 perhaps. A41 yields 13 2 1; we could insert "28" between "2" and "1". That would be 13 2 28 1, matching target positions 32-35. Then preceding 24 matches position 31. Good.

Alternatively, A16 -> 25 (rule_18). A42 -> 12 2 35 1 (rule_55). Actually A42 currently yields 12 2 35 1. That's a perfect match for positions 6-10: 25 12 2 35 1? No, that's 25 (A16) then A42 yields 12 2 35 1. Good! So A40 using rule_53 (A16 A42) yields exactly 25 12 2 35 1. That matches positions 6-10 exactly! Let's verify: A16 -> 25; A42 -> 12 2 35 1 is rule_55: A42 -> 12 2 35 1. Actually original rule_55 is A42 -> 12 2 35 1 (assuming that's correct). Let's confirm rule 55: "A42 -> 12 2 35 1". Yes. So A16 A42 yields 25 12 2 35 1, exactly positions 6-10.

Thus we choose A40 = A16 A42 via rule_53. Great, no insert needed there. So A40 yields positions 6-10.

Now A37 must produce the next substring starting at position 11: 26 7 2 34 1 9 8 32 33 31 17 16 21 30 20 19 ... but hold on, we need to examine how A37 could produce that.

Recall A37 alternatives: A16 A38 (rule_48) OR A17 A39 (rule_49). A16 is 25, not 26, so not relevant for 26 start. A17 is 26. So we should use A37 -> A17 A39, since target at pos11 starts with 26. Let's see A17 yields 26. A39 yields 7 2 34 1. That's rule_51: "A39 -> 7 2 34 1". That matches next four numbers: 7,2,34,1 (positions 12-15). So A37 produces exactly 26 7 2 34 1. Perfect. No insertion needed. So A37 yields positions 11-15.

Thus far we have A1 expansions: A43 -> 11 10 2 36 1, A40 -> 25 12 2 35 1, A37 -> 26 7 2 34 1, correct.

Now A33 must produce the substring starting at position 16: we have now consumed positions 1-15. Next is positions 16 onward: Let's list from pos16:

pos16: 9
pos17: 8
pos18: 32
pos19: 33
pos20: 31
pos21: 17
pos22: 16
pos23: 21
pos24: 30
pos25: 20
pos26: 19
pos27: 18
pos28: 21
pos29: 29
pos30: 20
pos31: 24
pos32: 13
pos33: 2
pos34: 28
pos35: 1
pos36: 15
pos37: 14
pos38: 21
pos39: 27
pos40: 20
pos41: 23
pos42: 3
pos43: 21
pos44: 22
pos45: 20
pos46: 6
pos47: 5
pos48: 2
pos49: 4
pos50: 1

Thus remaining after A33 etc.

Now A33 has three alternatives:

- A5 A34 (rule_42)
- A6 A35 (rule_43)
- A8 A36 (rule_44)

Thus A33 yields either:

Option1: A5 (9) + A34 (8 33 31) => 9 8 33 31.

Option2: A6 (11) + A35 (10 32 31) => 11 10 32 31.

Option3: A8 (15) + A36 (14) => 15 14.

Our remaining target begins with 9 8 32 33 31 ... However note that Option1 yields 9 8 33 31, but target after that is 9 8 32 33 31. The sequence is 9,8,32,33,31.

Thus we can use Option1 (A5 A34) to produce 9 8 33 31 but missing "32". The target has "32" inserted between 8 and 33 31 (so after 8 before 33). That suggests we need to insert terminal 32 into either A5's rule (A5 yields only 9) or into A34 (which yields 8 33 31). In A34 we could insert "32" after the 8 (which currently only yields 8 33 31). Actually A34 has RHS: 8 33 31. We could insert 32 between 8 and 33, making it "8 32 33 31". That would produce 9 (from A5) then 8 32 33 31 = 9 8 32 33 31, matching the target positions 16-20. That would require inserting one terminal "32". Since A34 currently has 8 33 31. Insert 32 after 8 => 2 insertions? Actually we insert a terminal, not a sequence. So one insertion.

Alternatively could insert "32" after A5? A5's rule yields 9; we could make it "9 8" but that would modify shape and cause duplicate. So best to add to A34.

Thus we choose A33 as A5 A34 (rule_42). We add a "32" after the 8 in rule_45 (A34). That yields A34 -> 8 32 33 31. We increment insert count by 1.

Now A33 yields "9 8 32 33 31". That's positions 16-20, matches target 9,8,32,33,31.

Thus A33 is now done.

Now next is A29. It must produce sequence starting at position 21: that's 17 16 21 30 20 19 18 21 29 20 24? Wait include 17 onward.

Positions:

21: 17
22: 16
23: 21
24: 30
25: 20
26: 19
27: 18
28: 21
29: 29
30: 20
31: 24
... So A29 must derive "17 16 21 30 20 19 18 21 29 20 24 ..." Possibly continues longer. Let's see possible expansions for A29:

Options:
- A8 A30 (rule_36) => 15 + (A30 yields 14 21 20) => 15 14 21 20. That's not match.
- A9 A31 (rule_37) => 17 + (A31 -> 16 30) => yields 17 16 30.
- A10 A32 (rule_38) => 19 + (A32 -> 18 21 20) => yields 19 18 21 20.

So only potential matching start "17" is Option2: A9 A31. A9 yields 17, A31 yields 16 30. So A29 using rule_37 yields "17 16 30". Our target at positions 21-23 is "17 16 21". Actually we have 17 16 21 30... So after 17 16 we have 21, then 30, not in order. So maybe we need to combine A31 rule to produce "16 21 30"? No, A31 currently yields "16 30". But we could insert "21" into A31's RHS.

Rule_40: A31 -> 16 30. Insert "21" between "16" and "30". That would give "16 21 30". That matches target's sequence "16 21 30". Perfect. So we can modify rule_40 to be 16 21 30 (one insertion of 21). Then A29 yields 17 (from A9) + 16 21 30 = 17 16 21 30. That matches positions 21-24.

But target still has after that "20 19 18 21 29 20 ...". So after "17 16 21 30" we have "20 19 18 21 29 20". That's likely the continuation from other nonterminals (maybe A24?). Actually A29's production includes only A9 and A31, so yields exactly 4 numbers? Actually after insertion, we have 4 numbers: 17, 16, 21, 30. No more numbers. So the rest must be derived by subsequent nonterminals: A24, A20, A18, A11, A2. Indeed at this point after A33 is done, we have A29 next, then A24, A20, A18, A11, A2.

Thus A24 must correspond to the next substring after A29's yields, i.e., starting at position 25 onward. Let's verify positions 25 onward.

We have after pos24 which is 30, sequence pos25: 20, pos26: 19, pos27: 18, pos28: 21, pos29: 29, pos30: 20, pos31: 24, pos32: 13, pos33: 2, pos34: 28, pos35: 1, pos36: 15, pos37: 14, pos38: 21, pos39: 27, pos40: 20, pos41: 23, pos42: 3, pos43: 21, pos44: 22, pos45: 20, pos46: 6, pos47: 5, pos48: 2, pos49: 4, pos50: 1.

Thus after A29 we have subsequence:

[20, 19, 18, 21, 29, 20, 24, 13, 2, 28, 1, 15, 14, 21, 27, 20, 23, 3, 21, 22, 20, 6, 5, 2, 4, 1] length 26.

Now A24 has four alternatives:

- A5 A25 (rule_28): yields 9 + (A25 -> 8) = 9 8.
- A8 A26 (rule_29): yields 15 + (A26 -> 13 2 28) = 15 13 2 28.
- A9 A27 (rule_30): yields 17 + (A27 -> 16) = 17 16.
- A10 A28 (rule_31): yields 19 + (A28 -> 18 21 29 20) = 19 18 21 29 20.

Observing target starting at position 25: 20 19 18 21 29 20. None of the A24 options directly produce 20 at start. But note A24 yields sequences of lengths 2 (A5 A25), 4 (A8 A26), 2 (A9 A27), or 5 (A10 A28: 19 18 21 29 20). The target after A31 yields "20 19 18 21 29 20". That's 6 numbers: 20, 19, 18, 21, 29, 20.

Possible approach: The sequence "19 18 21 29 20" matches A24's option A10 A28, which yields exactly that (since A10 yields 19; A28 yields 18 21 29 20). That matches positions 26-30 (starting at pos26?), let's double-check: target pos26: 19, pos27: 18, pos28: 21, pos29: 29, pos30: 20. Indeed those match exactly. The preceding "20" (position 25) is not part of that; maybe belongs to previous nonterminal (i.e., A20 or something). However A20 is placed after A24 according to A1. Actually A1 sequence: ... A29 A24 A20 A18 A11 A2. So after A24 we have A20, then A18, etc. So the "20" before "19 18 21 29 20" is part of A20's output (or perhaps inserted). So A24 should produce "19 18 21 29 20", which matches its option A10 A28 (rule_31). Good.

Thus we will select A24 -> A10 A28 (rule_31). That yields "19 18 21 29 20". There are no missing symbols inside. So A24 yields positions 26-30. However we need A24 to start at position 25? Actually we need to align the production boundaries precisely. Let's compute positions:

We have so far:

- A43 yields pos1-5: 11 10 2 36 1
- A40 yields pos6-10: 25 12 2 35 1
- A37 yields pos11-15: 26 7 2 34 1
- A33 yields pos16-20: 9 8 32 33 31
- A29 yields pos21-24: 17 16 21 30

Now the next nonterminal is A24, which we plan to yield pos25-??. Let's see target again with positions:

1: 11
2: 10
3: 2
4: 36
5: 1

6: 25
7: 12
8: 2
9: 35
10: 1

11: 26
12: 7
13: 2
14: 34
15: 1

16: 9
17: 8
18: 32
19: 33
20: 31

21: 17
22: 16
23: 21
24: 30

25: 20
26: 19
27: 18
28: 21
29: 29
30: 20

31: 24
32: 13
33: 2
34: 28
35: 1

36: 15
37: 14
38: 21
39: 27
40: 20

41: 23
42: 3
43: 21
44: 22
45: 20

46: 6
47: 5
48: 2
49: 4
50: 1

Thus the segment "20 19 18 21 29 20" begins at pos25 (20) then pos26 (19) etc. However A24 yields "19 18 21 29 20". There's a missing "20" at start (position 25). That "20" could be produced by A20 (the next nonterminal) that comes after A24. But A20 comes after A24: the order is A24 then A20. So cannot produce a prefix preceding A24. Wait the order in A1 is A43 A40 A37 A33 A29 A24 A20 A18 A11 A2. So yes, after A29 we have A24, then A20, then A18, etc. So the "20" at position 25 must be part of A24's output if it's the first symbol after A29 (which ends at position 24). Since A24's output starts at position 25. Therefore A24 must produce "20" as its first terminal, then the rest. But A24's productions currently start with either 9, 15, 17, 19. No production starts with 20. So we need to insert a "20" somewhere in an A24 production.

Thus we could choose any of the alternatives for A24 and insert "20" somewhere. Which is best? The most similar to target subsequence after 20 is "19 18 21 29 20". So A24 -> A10 A28 yields "19 18 21 29 20". That's only missing the initial 20. So we could insert a 20 before A10 (i.e., at start of RHS). So rule_31: A24 -> A10 A28 originally. We could change it to "20 A10 A28" i.e., produce a terminal 20 before the nonterminal A10. However the rule operates as A24 -> A10 A28; we may insert any terminals anywhere in RHS: before A10, between A10 and A28, after A28. So we can insert "20" before A10, making RHS "20 A10 A28". That yields "20 19 18 21 29 20". That matches exactly positions 25-30.

Thus we insert one terminal "20" at the beginning of rule_31. That adds 1 insertion.

Thus A24 yields the correct six symbols.

Now after that, the next nonterminal is A20.

Now the target after position 30 is at position 31: 24 13 2 28 1 15 14 21 27 20 23 3 21 22 20 6 5 2 4 1.

Thus A20 must produce substring "24 13 2 28 1 15 14 21 27 20 23 3 21 22 20 6 5 2 4 1". That's 20 symbols.

Recall A20 productions:

- rule_22: A20 -> A13 A21
- rule_23: A20 -> A14 A22
- rule_24: A20 -> A16 A23

Thus we need to decide which yields matches the pattern:

Consider each:

- A13 yields 23. So A13 A21 yields 23 then A21 yields 3 1 => 23 3 1. That's not near start.

- A14 yields 24. A22 yields 13 2 28. So A14 A22 yields 24 13 2 28. That's exactly the first four symbols of A20's target segment: 24 13 2 28. Then after that we have "1 15 14 21 27 20 23 3 21 22 20 6 5 2 4 1". So it seems A20 may produce 24 13 2 28 1 followed by something else. Currently, A14 A22 yields only 24 13 2 28. It lacks the subsequent "1". We could insert "1" after A22. That's a straightforward insertion: in rule_23 (A20 -> A14 A22), we could insert "1" after A22: i.e., "A14 A22 1". That would yield 24 13 2 28 1. Good.

Now after that, the remaining substring for later nonterminals: "15 14 21 27 20 23 3 21 22 20 6 5 2 4 1". That's 15 symbols.

The next nonterminal after A20 is A18. Then A11, then A2.

So A18 must produce segment beginning after the "1". In target, after the "1" (pos35) we have "15". So A18 must produce "15 ..." perhaps plus more.

Recall A18 -> A8 A19 (rule_20). A8 yields 15; A19 yields 14 21 27. So A18 yields 15 14 21 27. That's four symbols: 15 14 21 27. The target after pos35 is "15 14 21 27 20 23 3 21 22 20 6 5 2 4 1". Indeed after the "1", we have "15 14 21 27". Then "20 23 3 ..." So we need A18 to produce 15 14 21 27 and then something else? Actually A18 yields exactly those four symbols. Then the next part "20 23 3 21 22 20 6 5 2 4 1" will be produced by A11 and A2.

Thus A18 is fine as is: rule_20: A18 -> A8 A19; A8 ->15; A19->14 21 27 (rule_21). So yields correct 15 14 21 27.

Thus we need no insertion for A18.

Now A11 next.

Target after the 27 is at pos40: 20, then 23 3 21 22 20 6 5 2 4 1. Actually let's see exactly: After 27 (pos39), position 40 is 20, position 41 is 23, 42 is 3, 43 is 21, 44 is 22, 45 is 20, then 46-50: 6 5 2 4 1.

Thus A11 must generate substring: 20 23 3 21 22 20 6 5 2 4 1? Wait we need to verify if A11 returns 20? Let's check A11 productions:

- rule_12: A11 -> A13 A12
- rule_13: A11 -> A16 A15

A13 yields 23. A12 yields 3 22 20. So A13 A12 yields 23 3 22 20.

That's close but we need maybe 23 3 21 22 20... Actually we have sequence "23 3 21 22 20". Our target: after "20" preceding A11 maybe we need to produce "23 3 21 22 20". Actually let's re-evaluate: The target after A18 is "20 23 3 21 22 20 6 5 2 4 1".

Thus A11 must generate "23 3 21 22 20"? Or perhaps A11 may generate "23 3 22 20" as per A13 A12, then we need to insert "21" after "3". Actually target after 23 is "3 21 22 20". So not "3 22 20". So we need to insert "21" after "3". That's one insertion.

Alternatively, A11 could use rule_13: A11 -> A16 A15. A16 yields 25; A15 yields 12 21 20 => gives 25 12 21 20. That's not matching our needed segment.

Thus we must use rule_12: A11 -> A13 A12. A13 yields 23; A12 yields 3 22 20. We need "23 3 21 22 20". So we need to insert "21" between "3" and "22". So we should insert terminal 21 into rule_14? Actually rule_14 defines A12: A12 -> 3 22 20. So we can insert "21" after the 3. So Modified rule_14: A12 -> 3 21 22 20. That adds 1 insertion (terminal 21). That yields A12 = 3 21 22 20. So A13 A12 yields 23 3 21 22 20. Perfect.

Thus A11 yields 23 3 21 22 20. However we need preceding "20" before A11? Actually before A11 there is A18 which yields "... 27". Right after that, target has "20". So that "20" may be left over from something else, maybe A20? No, A20's output we already accounted for up to its trailing "1". Then A18 yields 15 14 21 27. After that, target has "20". So after A18, there is a "20". That must be generated by something else prior to A11 (maybe an extra terminal inserted into A18's production?). However A18 yields exactly "15 14 21 27". It doesn't produce a 20 afterward. The "20" we need appears immediately after "27". Could we insert "20" after A19 in rule_20? That is: A18 -> A8 A19. So we could add a terminal "20" after A19: A18 -> A8 A19 20 . That would produce 15 14 21 27 20. That matches target after A18: "15 14 21 27 20". Then A11 yields "23 3 21 22 20". So overall we have "15 14 21 27 20 23 3 21 22 20". That matches positions 36-45 exactly. Good.

Thus we need to insert a "20" terminal after A19 in rule_20.

Thus A18 yields correct segment.

Now A2 final must produce suffix "6 5 2 4 1". Currently A2 can produce "6 5 2" or "15 14 2". We want "6 5 2 4 1". So as earlier we need to insert "4 1" into A3 (or A2) or A3. The easiest: modify A3 (rule_4: A3 -> 5 2) to add "4 1" after "2". So A3 becomes "5 2 4 1". That's 2 insertions.

Thus total insertions counted:

- Insert "36" into rule_61 (A45). (1)
- Insert "32" into rule_45 (A34). (1)
- Insert "21" into rule_40 (A31). (1)
- Insert "20" at the beginning of rule_31 (A24). (1)
- Insert "20" before A10 in rule_31? Actually we inserted "20" at start: yes, 1 insertion.
- Insert "20" after A19 in rule_20 (A18). (1)
- Insert "21" into rule_14 (A12). (1)
- Insert "4 1" into rule_4 (A3) (2)
Total: 1+1+1+1+1+1+1+2 = 9 insertions.

Did we consider any extra insertion for rule_45 (A34) we added only 32. That's 1.

Double-check count: the modifications are:

1. Rule_61 (A45): add 36 (1)
2. Rule_45 (A34): add 32 (1)
3. Rule_40 (A31): add 21 (1)
4. Rule_31 (A24): add leading 20 (1)
5. Rule_20 (A18): add trailing 20 (1)
6. Rule_14 (A12): add 21 (1)
7. Rule_4 (A3): add 4 1 (2)
-> total 8? Wait 1+1+1+1+1+1+2 = 8. Did we count 9? I also considered maybe inserting something else for A20? We inserted "1" after rule_23 (A20). Did we add that? Yes we needed to insert "1" after A22 to get trailing 1. Let's verify: A20 rule_23: A20 -> A14 A22. We need to insert terminal "1" after A22 (or after rule). So we insert "1" at end of RHS: A20 -> A14 A22 1. That's another insertion. So add that. That's 1 insertion.

Thus now total insertions:

- Insert "36" in rule_61 (A45)
- Insert "32" in rule_45 (A34)
- Insert "21" in rule_40 (A31)
- Insert leading "20" in rule_31 (A24)
- Insert "1" at end of rule_23 (A20)
- Insert trailing "20" after A19 in rule_20 (A18)
- Insert "21" in rule_14 (A12)
- Insert "4 1" into rule_4 (A3) (two insertions)
Total = 1+1+1+1+1+1+1+2 = 9 insertions.

Thus total 9.

Now we need to verify that all target segments are covered:

Let's rewrite the final grammar derivation stepwise for sanity:

- A1 -> A43 A40 A37 A33 A29 A24 A20 A18 A11 A2

Expansions:

A43 -> A6 A45 (choose rule_57)
- A6 -> 11
- A45 -> 10 2 36 1 (modified)
Thus A43 yields: 11 10 2 36 1

A40 -> A16 A42 (rule_53)
- A16 -> 25
- A42 -> 12 2 35 1 (as is)
Thus A40 yields: 25 12 2 35 1

A37 -> A17 A39 (rule_49)
- A17 -> 26
- A39 -> 7 2 34 1
Thus A37 yields: 26 7 2 34 1

A33 -> A5 A34 (rule_42)
- A5 -> 9
- A34 -> 8 32 33 31 (modified; original 8 33 31, added 32 after 8)
Thus A33 yields: 9 8 32 33 31

A29 -> A9 A31 (rule_37)
- A9 -> 17
- A31 -> 16 21 30 (modified; original 16 30, added 21)
Thus A29 yields: 17 16 21 30

A24 -> 20 A10 A28 (modified rule_31)
- first terminal 20 inserted
- A10 -> 19
- A28 -> 18 21 29 20
Thus yields: 20 19 18 21 29 20

A20 -> A14 A22 1 (modified rule_23)
- A14 -> 24
- A22 -> 13 2 28
- terminal 1 after
Thus yields: 24 13 2 28 1

A18 -> A8 A19 20 (modified rule_20)
- A8 -> 15
- A19 -> 14 21 27
- added 20
Thus yields: 15 14 21 27 20

A11 -> A13 A12 (rule_12)
- A13 -> 23
- A12 -> 3 21 22 20 (modified, original 3 22 20, inserted 21)
Thus yields: 23 3 21 22 20

A2 -> A4 A3 (rule_2)
- A4 -> 6
- A3 -> 5 2 4 1 (modified)
Thus yields: 6 5 2 4 1

Now concatenating all yields:

A43: 11 10 2 36 1
A40: 25 12 2 35 1
A37: 26 7 2 34 1
A33: 9 8 32 33 31
A29: 17 16 21 30
A24: 20 19 18 21 29 20
A20: 24 13 2 28 1
A18: 15 14 21 27 20
A11: 23 3 21 22 20
A2: 6 5 2 4 1

Let's list them linearly:

1-5: 11 10 2 36 1 (OK)
6-10: 25 12 2 35 1 (OK)
11-15: 26 7 2 34 1 (OK)
16-20: 9 8 32 33 31 (OK)
21-24: 17 16 21 30 (OK)
25-30: 20 19 18 21 29 20 (positions 25-30)
31-35: 24 13 2 28 1 (OK)
36-40: 15 14 21 27 20 (OK)
41-45: 23 3 21 22 20 (OK)
46-50: 6 5 2 4 1 (OK)

Now compare with target:

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

Exactly matches! All good.

Thus we have a correct derivation with 9 inserted terminals.

Now we need to output updated production rules lines for the rules we modified, with rule numbers and new RHS.

We must write:

Updated production rules:
rule_4: A3 -> 5 2 4 1
rule_14: A12 -> 3 21 22 20
rule_20: A18 -> A8 A19 20
rule_23: A20 -> A14 A22 1
rule_31: A24 -> 20 A10 A28
rule_40: A31 -> 16 21 30
rule_45: A34 -> 8 32 33 31
rule_61: A45 -> 10 2 36 1

And also rule_4 we already have; but note we inserted into rule_4 two symbols, but we just show final form. Also rule_45 and rule_61 we inserted respectively.

Also rule_20 is A18. In original rule_20: A18 -> A8 A19. We added "20". So new form: "A18 -> A8 A19 20".

Check rule_23: original "A20 -> A14 A22". New: "A20 -> A14 A22 1". Insert after A22.

Check rule_31: original "A24 -> A10 A28". New: "A24 -> 20 A10 A28". Insert before A10.

Check rule_40: original "A31 -> 16 30". New: "A31 -> 16 21 30".

Check rule_45: original "A34 -> 8 33 31". New: "A34 -> 8 32 33 31".

Check rule_61: original "A45 -> 10 2 1". New: "A45 -> 10 2 36 1". Good.

Check rule_4: original "A3 -> 5 2". New: "A3 -> 5 2 4 1". Good.

Check rule_14: original "A12 -> 3 22 20". New: "A12 -> 3 21 22 20". (We inserted 21 after 3.)

Now count updates: 8 rules changed actually. Wait we have rule_55 for A42 originally had 12 2 35 1; we didn't modify it. That's fine. Did we need to modify other rules for insertion of A41's "1"? Actually we inserted "1" at end of A20's rule (rule_23). So we reflect that.

Thus we have modified 8 rules: rule_4, rule_14, rule_20, rule_23, rule_31, rule_40, rule_45, rule_61. That's 8 updates, corresponding to 9 inserted terminals (some rules have 2 inserted terminals). Actually rule_4 had 2 inserted terminals, others 1 each.

Check if any other modifications needed: A45 now yields "10 2 36 1". This matches prefix. That is fine.

Everything else aligns.

Now we must produce exactly the output request: a new line with "Updated production rules:", then each updated rule in separate line with proper rule identifier and new RHS. No extra text after updated rules.

Thus final answer format:

Updated production rules:
rule_4: A3 -> 5 2 4 1
rule_14: A12 -> 3 21 22 20
rule_20: A18 -> A8 A19 20
rule_23: A20 -> A14 A22 1
rule_31: A24 -> 20 A10 A28
rule_40: A31 -> 16 21 30
rule_45: A34 -> 8 32 33 31
rule_61: A45 -> 10 2 36 1

Make sure we keep them in the original order? Not required but can be any order.

Thus answer.Updated production rules:
rule_4: A3 -> 5 2 4 1
rule_14: A12 -> 3 21 22 20
rule_20: A18 -> A8 A19 20
rule_23: A20 -> A14 A22 1
rule_31: A24 -> 20 A10 A28
rule_40: A31 -> 16 21 30
rule_45: A34 -> 8 32 33 31
rule_61: A45 -> 10 2 36 1