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 29 terminals, ranging from 1 to 29.
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 A28 A20 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A11
rule_4: A2 -> 1 A7 A12 3
rule_5: A2 -> A9 A7 4 3
rule_6: A2 -> A10 A7
rule_7: A2 -> A13 A7
rule_8: A2 -> A19 A7 4
rule_9: A3 -> 1
rule_10: A3 -> 1 A4
rule_11: A4 -> A6
rule_12: A5 -> 1
rule_13: A6 -> 1
rule_14: A6 -> 2
rule_15: A7 -> 1
rule_16: A7 -> 1 A8
rule_17: A8 -> A9
rule_18: A9 -> 1
rule_19: A9 -> 5
rule_20: A9 -> 6
rule_21: A9 -> 7
rule_22: A10 -> 1
rule_23: A10 -> 8
rule_24: A11 -> A5
rule_25: A11 -> A6
rule_26: A12 -> A9
rule_27: A13 -> A14
rule_28: A13 -> A16
rule_29: A14 -> A15
rule_30: A15 -> 1
rule_31: A16 -> A6 1
rule_32: A17 -> A18 1 9 10
rule_33: A18 -> 1
rule_34: A18 -> 11
rule_35: A19 -> A17
rule_36: A20 -> 1
rule_37: A20 -> A7 A21 A25 12
rule_38: A20 -> A10 A21
rule_39: A20 -> A26 A21 12
rule_40: A21 -> 1
rule_41: A21 -> A23 14 A22
rule_42: A22 -> A9 15
rule_43: A22 -> A10
rule_44: A23 -> 1
rule_45: A23 -> 1 16 A24
rule_46: A24 -> A9
rule_47: A25 -> A10
rule_48: A26 -> A27 8
rule_49: A27 -> A6 A3 18
rule_50: A28 -> 1
rule_51: A28 -> 1 A29 A34
rule_52: A28 -> A31 A29 24
rule_53: A28 -> A35 A29 24
rule_54: A29 -> 1
rule_55: A29 -> 1 22 A33
rule_56: A29 -> 1 A30
rule_57: A30 -> A32
rule_58: A31 -> 1
rule_59: A31 -> 21
rule_60: A32 -> 1
rule_61: A33 -> A31 23
rule_62: A33 -> A32
rule_63: A34 -> A31 24
rule_64: A35 -> A36
rule_65: A36 -> A9 A23 15
rule_66: A37 -> 1
rule_67: A37 -> 1 A38 A43
rule_68: A37 -> A40 A38 26 25
rule_69: A37 -> A44 A38 25
rule_70: A38 -> 1
rule_71: A38 -> 1 28 A42
rule_72: A38 -> 1 A39
rule_73: A39 -> A41
rule_74: A40 -> 1
rule_75: A40 -> 27
rule_76: A41 -> 1
rule_77: A42 -> A40 29
rule_78: A42 -> A41
rule_79: A43 -> A40
rule_80: A44 -> A45 27
rule_81: A45 -> A9 17

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