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 35 terminals, ranging from 1 to 35.
Each terminal symbol occurs in the target sequence at least once.

There are 45 non-terminal symbols, ranging from A1 to A45.

Production rules:
rule_1: A1 -> A37 A31 A22 A15 A9 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A6 2
rule_4: A2 -> A5 A3 3
rule_5: A3 -> 1
rule_6: A3 -> 1 A4
rule_7: A4 -> A5
rule_8: A5 -> 1
rule_9: A5 -> 4
rule_10: A6 -> A5 3
rule_11: A7 -> 1
rule_12: A7 -> 6
rule_13: A8 -> 1
rule_14: A9 -> 1
rule_15: A9 -> 1 A10 A13 8
rule_16: A9 -> A11 A10
rule_17: A9 -> A12 A10
rule_18: A9 -> A14 11 A10
rule_19: A10 -> 1
rule_20: A11 -> 1
rule_21: A11 -> 11
rule_22: A11 -> 12
rule_23: A12 -> 1
rule_24: A13 -> A11
rule_25: A13 -> 13
rule_26: A14 -> A11 10
rule_27: A15 -> 1
rule_28: A15 -> 1 A16 A20 14
rule_29: A15 -> A18 A16 20
rule_30: A15 -> A19 A16 14
rule_31: A15 -> A21 A16 14
rule_32: A16 -> 1
rule_33: A16 -> 1 A17
rule_34: A17 -> A18 15
rule_35: A17 -> A19
rule_36: A18 -> 1
rule_37: A18 -> 16
rule_38: A18 -> 17
rule_39: A18 -> 18
rule_40: A19 -> 1
rule_41: A19 -> 19
rule_42: A20 -> A18 20
rule_43: A20 -> 21
rule_44: A21 -> A18 1 21
rule_45: A22 -> 1
rule_46: A22 -> 1 A23 A26 22
rule_47: A22 -> A25 A23 23
rule_48: A22 -> A27 A23
rule_49: A22 -> A29 A23
rule_50: A23 -> 1
rule_51: A24 -> 1
rule_52: A25 -> 1
rule_53: A25 -> 24
rule_54: A26 -> A25 23
rule_55: A27 -> A28
rule_56: A28 -> A24
rule_57: A29 -> A30 24
rule_58: A30 -> A19 26
rule_59: A31 -> 1
rule_60: A31 -> 1 A32 A34 27
rule_61: A31 -> A33 A32 28 27
rule_62: A32 -> 1
rule_63: A33 -> 1
rule_64: A33 -> 29
rule_65: A34 -> A33
rule_66: A35 -> 1
rule_67: A35 -> 1 7 A36
rule_68: A36 -> A7
rule_69: A36 -> A8
rule_70: A37 -> 1
rule_71: A37 -> 1 A38 A41 32
rule_72: A37 -> A18 A38 33 32
rule_73: A37 -> A19 A38 32
rule_74: A37 -> A42 A38 33 32
rule_75: A38 -> 1
rule_76: A38 -> 1 A39
rule_77: A38 -> 1 A40
rule_78: A39 -> A18
rule_79: A39 -> A19
rule_80: A40 -> A19
rule_81: A41 -> A18 33
rule_82: A42 -> A43
rule_83: A43 -> A7 A44 34
rule_84: A44 -> 1
rule_85: A44 -> A35 30 A45
rule_86: A45 -> A7 31

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