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 41 non-terminal symbols, ranging from A1 to A41.

Production rules:
rule_1: A1 -> A32 A23 A15 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A6 A9 3
rule_4: A2 -> A5 A3
rule_5: A2 -> A7 A6 3
rule_6: A2 -> A8 A6 3
rule_7: A2 -> A11 A3
rule_8: A2 -> A14 A6 3
rule_9: A3 -> 1
rule_10: A3 -> 1 A4
rule_11: A4 -> A5
rule_12: A5 -> 1
rule_13: A5 -> 2
rule_14: A6 -> 1
rule_15: A7 -> 1
rule_16: A7 -> 5
rule_17: A7 -> 6
rule_18: A7 -> 7
rule_19: A8 -> 1
rule_20: A9 -> A7 4
rule_21: A10 -> A8
rule_22: A11 -> A10
rule_23: A12 -> A13 1 8 9
rule_24: A13 -> 1
rule_25: A13 -> 10
rule_26: A14 -> A12
rule_27: A15 -> 1
rule_28: A15 -> A6 A16 A20
rule_29: A15 -> A7 A16 16
rule_30: A15 -> A21 A16
rule_31: A16 -> 1
rule_32: A16 -> A18 12 A17
rule_33: A17 -> A7 13
rule_34: A18 -> 1
rule_35: A18 -> 1 A19
rule_36: A19 -> A7 15
rule_37: A19 -> A8
rule_38: A20 -> A7
rule_39: A20 -> A8
rule_40: A21 -> A22
rule_41: A22 -> A5 A3 17
rule_42: A23 -> 1
rule_43: A23 -> 1 A24 A29
rule_44: A23 -> A26 A24 24 19
rule_45: A23 -> A30 A24 19
rule_46: A24 -> 1
rule_47: A24 -> 1 A25
rule_48: A24 -> 1 A28
rule_49: A25 -> A26
rule_50: A26 -> 1
rule_51: A26 -> 20
rule_52: A27 -> 1
rule_53: A27 -> 21
rule_54: A28 -> A27
rule_55: A29 -> A26
rule_56: A29 -> A27
rule_57: A30 -> A31 20
rule_58: A31 -> A7 A18 13
rule_59: A32 -> 1
rule_60: A32 -> 1 A33 A37
rule_61: A32 -> A34 A33 26 25
rule_62: A32 -> A35 A33
rule_63: A32 -> A38 A33 26 25
rule_64: A32 -> A40 A33 25
rule_65: A33 -> 1
rule_66: A33 -> 1 A36
rule_67: A34 -> 1
rule_68: A34 -> 27
rule_69: A35 -> 1
rule_70: A36 -> A34 29
rule_71: A36 -> A35
rule_72: A37 -> A34 26
rule_73: A38 -> A39
rule_74: A39 -> A7 1 15
rule_75: A40 -> A41
rule_76: A41 -> A35 1

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