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 69 non-terminal symbols, ranging from A1 to A69.

Production rules:
rule_1: A1 -> A62 A54 A37 A18 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A8
rule_4: A2 -> A5 A3 A7
rule_5: A2 -> A14 A3 A13
rule_6: A3 -> 1
rule_7: A4 -> 1
rule_8: A5 -> 1
rule_9: A5 -> 3
rule_10: A6 -> 1
rule_11: A7 -> 4 2
rule_12: A8 -> A9
rule_13: A9 -> A4
rule_14: A9 -> A5 4
rule_15: A9 -> A6
rule_16: A10 -> 1
rule_17: A11 -> 1
rule_18: A12 -> 1
rule_19: A12 -> 6
rule_20: A13 -> 4 2
rule_21: A14 -> A15 3
rule_22: A15 -> A16 1 A17
rule_23: A16 -> 1
rule_24: A16 -> 8
rule_25: A17 -> 5 7
rule_26: A18 -> 1
rule_27: A18 -> 1 A19 A26 9
rule_28: A18 -> A22 A19 9
rule_29: A18 -> A24 A19
rule_30: A18 -> A28 A19 9
rule_31: A18 -> A32 A19 10
rule_32: A19 -> 1
rule_33: A19 -> 1 A20
rule_34: A19 -> 1 A25
rule_35: A20 -> A21
rule_36: A21 -> 1
rule_37: A22 -> 1
rule_38: A22 -> 11
rule_39: A23 -> 1
rule_40: A24 -> 1
rule_41: A24 -> 12
rule_42: A25 -> A21
rule_43: A25 -> A22 14
rule_44: A25 -> A23
rule_45: A26 -> A21
rule_46: A26 -> A24 10
rule_47: A27 -> 1
rule_48: A27 -> 15
rule_49: A28 -> A29
rule_50: A28 -> A31
rule_51: A29 -> A30 1
rule_52: A30 -> 1
rule_53: A30 -> 17
rule_54: A31 -> A23 1
rule_55: A32 -> A33 12
rule_56: A32 -> A35
rule_57: A33 -> A34
rule_58: A33 -> A34 1 18
rule_59: A34 -> 1
rule_60: A34 -> 19
rule_61: A35 -> A24
rule_62: A36 -> 1
rule_63: A37 -> 1
rule_64: A37 -> 1 A38 A46
rule_65: A37 -> A16 A38 A45
rule_66: A37 -> A52 A38 A51
rule_67: A38 -> 1
rule_68: A38 -> A41 A39
rule_69: A39 -> 21 A40
rule_70: A40 -> A12 22
rule_71: A40 -> A16
rule_72: A41 -> 1
rule_73: A41 -> A44 A42
rule_74: A42 -> 23 A43
rule_75: A43 -> A10
rule_76: A43 -> A16
rule_77: A44 -> 1
rule_78: A45 -> 25 20
rule_79: A46 -> A47 20
rule_80: A46 -> A48
rule_81: A47 -> A11
rule_82: A47 -> A16 25
rule_83: A48 -> A10
rule_84: A48 -> A11
rule_85: A48 -> A16
rule_86: A49 -> 1
rule_87: A49 -> 1 16 A50
rule_88: A50 -> A27
rule_89: A50 -> A34 18
rule_90: A51 -> 25
rule_91: A52 -> A53 8
rule_92: A53 -> A30 A49
rule_93: A53 -> A30 A49 27
rule_94: A54 -> 1
rule_95: A54 -> 1 A55 A57
rule_96: A54 -> A27 A55 29
rule_97: A54 -> A34 A55
rule_98: A54 -> A36 A55
rule_99: A54 -> A59 A55 29 28
rule_100: A54 -> A61 A55
rule_101: A55 -> 1
rule_102: A55 -> 1 A56
rule_103: A56 -> A27
rule_104: A56 -> A36
rule_105: A57 -> A27 29
rule_106: A57 -> A30
rule_107: A57 -> A36
rule_108: A58 -> A22 1
rule_109: A59 -> A58 15
rule_110: A60 -> A30
rule_111: A61 -> A60
rule_112: A62 -> 1
rule_113: A62 -> A34 A63
rule_114: A62 -> A55 A63 A66 30
rule_115: A62 -> A67 A63 31 30
rule_116: A63 -> 1
rule_117: A63 -> A49 26 A65
rule_118: A63 -> A55 A64
rule_119: A64 -> A27
rule_120: A64 -> A34
rule_121: A64 -> A36
rule_122: A65 -> A30 27
rule_123: A66 -> A34
rule_124: A67 -> A68 19
rule_125: A68 -> A10 A44 A69
rule_126: A69 -> 23 24

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