I have a context free grammar (CFG). The initial symbol is `A1`. The terminal symbols are just integers and the non-terminal symbols start with `A`. I also have a target sequence of terminal symbols which needs to be produced by the grammar. However, the CFG production rules have had some terminal symbols removed which may now render it impossible to produce the target sequence. Please attempt to repair the CFG so that the sequence can be produced. The only operations which can be made to repair the CFG are the insertion of terminal symbols anywhere in existing production rules. Deleting 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 that the CFG can produce the target sequence. Reason about the problem however you wish and write out whatever working out you want. 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.


"""
There are 36 terminals, ranging from 1 to 36.
Each terminal symbol occurs in the target sequence at least once.

There are 47 non-terminal symbols, ranging from A1 to A47.

Production rules:
rule_1: A1 -> A43 A40 A37 A33 A29 A24 A20 A18 A11 A2
rule_2: A2 -> A4 A3
rule_3: A2 -> A8 A7
rule_4: A3 -> 5 2
rule_5: A4 -> 6
rule_6: A5 -> 9
rule_7: A6 -> 11
rule_8: A7 -> 14 2
rule_9: A8 -> 15
rule_10: A9 -> 17
rule_11: A10 -> 19
rule_12: A11 -> A13 A12
rule_13: A11 -> A16 A15
rule_14: A12 -> 3 22 20
rule_15: A13 -> 23
rule_16: A14 -> 24
rule_17: A15 -> 12 21 20
rule_18: A16 -> 25
rule_19: A17 -> 26
rule_20: A18 -> A8 A19
rule_21: A19 -> 14 21 27
rule_22: A20 -> A13 A21
rule_23: A20 -> A14 A22
rule_24: A20 -> A16 A23
rule_25: A21 -> 3 1
rule_26: A22 -> 13 2 28
rule_27: A23 -> 12 2 1
rule_28: A24 -> A5 A25
rule_29: A24 -> A8 A26
rule_30: A24 -> A9 A27
rule_31: A24 -> A10 A28
rule_32: A25 -> 8
rule_33: A26 -> 14 20
rule_34: A27 -> 16
rule_35: A28 -> 18 21 29 20
rule_36: A29 -> A8 A30
rule_37: A29 -> A9 A31
rule_38: A29 -> A10 A32
rule_39: A30 -> 14 21 20
rule_40: A31 -> 16 30
rule_41: A32 -> 18 21 20
rule_42: A33 -> A5 A34
rule_43: A33 -> A6 A35
rule_44: A33 -> A8 A36
rule_45: A34 -> 8 33 31
rule_46: A35 -> 10 32 31
rule_47: A36 -> 14
rule_48: A37 -> A16 A38
rule_49: A37 -> A17 A39
rule_50: A38 -> 12 2 1
rule_51: A39 -> 7 2 34 1
rule_52: A40 -> A14 A41
rule_53: A40 -> A16 A42
rule_54: A41 -> 13 2 1
rule_55: A42 -> 12 2 35 1
rule_56: A43 -> A5 A44
rule_57: A43 -> A6 A45
rule_58: A43 -> A8 A46
rule_59: A43 -> A10 A47
rule_60: A44 -> 8 2 1
rule_61: A45 -> 10 2 1
rule_62: A46 -> 14 1
rule_63: A47 -> 18 1

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
"""