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 31 terminals, ranging from 1 to 31.
Each terminal symbol occurs in the target sequence at least once.

There are 62 non-terminal symbols, ranging from A1 to A62.

Production rules:
rule_1: A1 -> A53 A45 A29 A16 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A6
rule_4: A2 -> A4 A3 A5
rule_5: A2 -> A12 A3 A11
rule_6: A3 -> 1
rule_7: A4 -> 1
rule_8: A4 -> 3
rule_9: A5 -> 4 2
rule_10: A6 -> A7 2
rule_11: A7 -> A4 4
rule_12: A8 -> 1
rule_13: A9 -> 1
rule_14: A9 -> 6
rule_15: A10 -> 1
rule_16: A10 -> 7
rule_17: A11 -> 4
rule_18: A12 -> A13 3
rule_19: A13 -> A14 A15
rule_20: A14 -> 1
rule_21: A14 -> 9
rule_22: A15 -> 5 8
rule_23: A16 -> 1
rule_24: A16 -> 1 A17 A22 10
rule_25: A16 -> A18 A17
rule_26: A16 -> A19 A17 10
rule_27: A16 -> A20 A17 10
rule_28: A16 -> A26 A17 14 10
rule_29: A17 -> 1
rule_30: A17 -> 1 A21
rule_31: A18 -> 1
rule_32: A18 -> 11
rule_33: A19 -> 1
rule_34: A20 -> 1
rule_35: A21 -> A18
rule_36: A21 -> A19
rule_37: A21 -> A20
rule_38: A22 -> A18 14
rule_39: A23 -> 1
rule_40: A23 -> 15
rule_41: A24 -> 1
rule_42: A25 -> 1
rule_43: A26 -> A27 11
rule_44: A27 -> A28 1 16
rule_45: A28 -> 1
rule_46: A28 -> 18
rule_47: A28 -> 19
rule_48: A29 -> 1
rule_49: A29 -> 1 A31 A38
rule_50: A29 -> A9 A31 A30
rule_51: A29 -> A43 A31 A42
rule_52: A30 -> 21 20
rule_53: A31 -> 1
rule_54: A31 -> A34 A32
rule_55: A32 -> 22 A33
rule_56: A33 -> A10
rule_57: A33 -> A14 23
rule_58: A34 -> 1
rule_59: A34 -> A37 A35
rule_60: A35 -> 24 A36
rule_61: A36 -> A10 25
rule_62: A36 -> A14
rule_63: A37 -> 1
rule_64: A38 -> A39 20
rule_65: A39 -> A8
rule_66: A39 -> A9
rule_67: A39 -> A10
rule_68: A40 -> 1
rule_69: A40 -> 1 A41
rule_70: A41 -> A23
rule_71: A41 -> A28 17
rule_72: A42 -> 21 20
rule_73: A43 -> A44 6
rule_74: A44 -> A23 A40 26 27
rule_75: A44 -> A23 A40 27
rule_76: A45 -> 1
rule_77: A45 -> 1 A46 A48 28
rule_78: A45 -> A28 A46 29
rule_79: A45 -> A49 A46 29
rule_80: A45 -> A51 A46 28
rule_81: A46 -> 1
rule_82: A46 -> 1 A47
rule_83: A47 -> A24
rule_84: A48 -> A24
rule_85: A48 -> A28 29
rule_86: A49 -> A50 18
rule_87: A50 -> A18 1 12
rule_88: A51 -> A52
rule_89: A52 -> A25
rule_90: A53 -> 1
rule_91: A53 -> A24 A54 30
rule_92: A53 -> A28 A54 30
rule_93: A53 -> A46 A54 A56
rule_94: A53 -> A57 A54 30
rule_95: A53 -> A60 A54 30
rule_96: A54 -> 1
rule_97: A54 -> A40 26 A55
rule_98: A55 -> A23
rule_99: A55 -> A25
rule_100: A56 -> A23
rule_101: A56 -> A25
rule_102: A56 -> A28 31
rule_103: A57 -> A58
rule_104: A58 -> A14 A37 A59
rule_105: A59 -> 24
rule_106: A60 -> A61 19
rule_107: A61 -> A10 A37 A62
rule_108: A62 -> 24

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