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 50 non-terminal symbols, ranging from A1 to A50.

Production rules:
rule_1: A1 -> A43 A37 A28 A20 A17 A15 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A8 2
rule_4: A2 -> A5 A3
rule_5: A2 -> A6 A3 2
rule_6: A2 -> A13 A3 9 2
rule_7: A3 -> 1
rule_8: A3 -> 1 A4
rule_9: A3 -> 1 A7
rule_10: A4 -> A5 4
rule_11: A5 -> 1
rule_12: A5 -> 5
rule_13: A5 -> 6
rule_14: A6 -> 1
rule_15: A6 -> 7
rule_16: A6 -> 8
rule_17: A7 -> A6
rule_18: A8 -> A6
rule_19: A9 -> 1
rule_20: A9 -> 1 A10
rule_21: A10 -> A11
rule_22: A10 -> A12
rule_23: A11 -> 1
rule_24: A11 -> 12
rule_25: A12 -> 1
rule_26: A12 -> 13
rule_27: A13 -> A14
rule_28: A14 -> A12 A9 14 15
rule_29: A15 -> 1
rule_30: A16 -> 1
rule_31: A16 -> 16
rule_32: A17 -> 1
rule_33: A17 -> 1 A18 A19
rule_34: A18 -> 1
rule_35: A19 -> A11
rule_36: A20 -> 1
rule_37: A20 -> 1 A21 A24 19
rule_38: A20 -> A22 A21 19
rule_39: A20 -> A23 A21 19
rule_40: A20 -> A25 A21 19
rule_41: A21 -> 1
rule_42: A22 -> 1
rule_43: A22 -> 21
rule_44: A23 -> 1
rule_45: A24 -> A22 20
rule_46: A24 -> A23
rule_47: A25 -> A26
rule_48: A25 -> A27
rule_49: A26 -> A11 1 11
rule_50: A26 -> A11 10 11
rule_51: A27 -> A22
rule_52: A28 -> 1
rule_53: A28 -> 1 A29 A33 22
rule_54: A28 -> A30 A29 26
rule_55: A28 -> A34 A29 26 22
rule_56: A29 -> 1
rule_57: A30 -> 1
rule_58: A30 -> 23
rule_59: A30 -> 24
rule_60: A31 -> 1
rule_61: A31 -> 25
rule_62: A32 -> 1
rule_63: A33 -> A30 26
rule_64: A33 -> A31
rule_65: A34 -> A35 24
rule_66: A34 -> A36 23
rule_67: A35 -> A16 1
rule_68: A35 -> A16 17 18
rule_69: A36 -> A5 1
rule_70: A37 -> 1
rule_71: A37 -> A29 A38 A41 27
rule_72: A37 -> A30 A38
rule_73: A37 -> A42 A38 28 27
rule_74: A38 -> 1
rule_75: A39 -> 1
rule_76: A39 -> A32 29 A40
rule_77: A40 -> A5
rule_78: A41 -> A30
rule_79: A42 -> A5 A32 29 30 23
rule_80: A43 -> 1
rule_81: A43 -> A30 A44 31
rule_82: A43 -> A31 A44
rule_83: A43 -> A38 A44 A46 31
rule_84: A43 -> A47 A44 32 31
rule_85: A43 -> A49 A44 31
rule_86: A44 -> 1
rule_87: A44 -> 1 33 A45
rule_88: A45 -> A31
rule_89: A46 -> A30
rule_90: A46 -> A31
rule_91: A47 -> A48
rule_92: A48 -> A6 A39 35
rule_93: A48 -> A6 A39 36 35
rule_94: A49 -> A50 23
rule_95: A50 -> A5 A39

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