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 43 terminals, ranging from 1 to 43.
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 -> A2
rule_2: A2 -> A43
rule_3: A3 -> A4 A5
rule_4: A4 -> 6
rule_5: A5 -> 4 A6
rule_6: A5 -> 5 A7
rule_7: A6 -> 1
rule_8: A6 -> 2
rule_9: A7 -> 1
rule_10: A7 -> 2
rule_11: A7 -> 7
rule_12: A8 -> 8
rule_13: A8 -> A9 A10
rule_14: A9 -> 9
rule_15: A10 -> 4 A11
rule_16: A10 -> 5 A12
rule_17: A11 -> 1
rule_18: A11 -> 2
rule_19: A11 -> 10
rule_20: A12 -> 1
rule_21: A12 -> 2
rule_22: A13 -> 11
rule_23: A14 -> 12
rule_24: A15 -> A16 A17
rule_25: A16 -> 14
rule_26: A17 -> 16 A18
rule_27: A17 -> 17 A19
rule_28: A18 -> 1
rule_29: A18 -> 2
rule_30: A18 -> 15
rule_31: A19 -> 1
rule_32: A19 -> 2
rule_33: A20 -> A21 A22
rule_34: A21 -> 18
rule_35: A22 -> 16 A23
rule_36: A22 -> 17 A24
rule_37: A23 -> 1
rule_38: A23 -> 2
rule_39: A24 -> 1
rule_40: A24 -> 2
rule_41: A25 -> 19
rule_42: A26 -> 20 21
rule_43: A27 -> A28 A29
rule_44: A28 -> 23
rule_45: A29 -> 22 A30
rule_46: A30 -> 1
rule_47: A30 -> 2
rule_48: A31 -> 24
rule_49: A32 -> 3 A33
rule_50: A33 -> 1
rule_51: A33 -> 2
rule_52: A33 -> 25
rule_53: A34 -> A35 A32
rule_54: A34 -> A36 A14 A3 A13
rule_55: A35 -> 26
rule_56: A36 -> 27 A8
rule_57: A37 -> A38 A26 A15 A25
rule_58: A38 -> 28 A20
rule_59: A39 -> 29 A34 30
rule_60: A40 -> 31 32
rule_61: A41 -> 33 A34 34
rule_62: A42 -> 35 A37
rule_63: A43 -> 43 37 A44 A46
rule_64: A43 -> 43 38 37 A46 A44
rule_65: A44 -> A45 A42 A3 A41
rule_66: A45 -> 41 A8
rule_67: A46 -> A47 A40 A27 A39
rule_68: A47 -> 42 A31

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