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 30 terminals, ranging from 1 to 30.
Each terminal symbol occurs in the target sequence at least once.

There are 82 non-terminal symbols, ranging from A1 to A82.

Production rules:
rule_1: A1 -> A73 A57 A51 A41 A32 A18 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A15
rule_4: A3 -> 1
rule_5: A3 -> A10 A8
rule_6: A4 -> 1
rule_7: A5 -> 1
rule_8: A5 -> 3
rule_9: A5 -> 4
rule_10: A6 -> 1
rule_11: A7 -> 1
rule_12: A8 -> 5 A9
rule_13: A9 -> A4
rule_14: A9 -> A5 6
rule_15: A10 -> 1
rule_16: A10 -> A13 A11
rule_17: A11 -> 7 A12
rule_18: A12 -> A4 8
rule_19: A12 -> A5
rule_20: A12 -> A7
rule_21: A13 -> 1
rule_22: A14 -> 1
rule_23: A15 -> A16
rule_24: A15 -> A17
rule_25: A16 -> A6
rule_26: A16 -> A7
rule_27: A16 -> 10
rule_28: A17 -> A4
rule_29: A17 -> A5
rule_30: A17 -> A7
rule_31: A18 -> 1
rule_32: A18 -> 1 A19 A25
rule_33: A18 -> A20 A19 A24
rule_34: A18 -> A31 A19 A30
rule_35: A19 -> 1
rule_36: A20 -> 1
rule_37: A20 -> 12
rule_38: A20 -> 13
rule_39: A21 -> 1
rule_40: A21 -> 14
rule_41: A22 -> 1
rule_42: A23 -> 1
rule_43: A23 -> 15
rule_44: A24 -> 16 11
rule_45: A25 -> A26 11
rule_46: A25 -> A27
rule_47: A26 -> A20 16
rule_48: A26 -> A22
rule_49: A27 -> A20
rule_50: A27 -> A21
rule_51: A27 -> A22
rule_52: A28 -> A5 A29
rule_53: A29 -> 17 9
rule_54: A30 -> 16
rule_55: A31 -> A28 12
rule_56: A32 -> 1
rule_57: A32 -> A14 A13 A36
rule_58: A32 -> A19 A33 A38
rule_59: A32 -> A23 A33 A35
rule_60: A33 -> 1
rule_61: A34 -> A20
rule_62: A34 -> A21
rule_63: A34 -> A23 19
rule_64: A35 -> 20 18
rule_65: A36 -> A37
rule_66: A37 -> A5
rule_67: A37 -> A6
rule_68: A37 -> A7
rule_69: A38 -> A39
rule_70: A38 -> A40 18
rule_71: A39 -> A20
rule_72: A39 -> 21
rule_73: A39 -> A22
rule_74: A40 -> A20
rule_75: A40 -> A23
rule_76: A41 -> 1
rule_77: A41 -> A21 A43 A42
rule_78: A41 -> A33 A43 A45
rule_79: A41 -> A48 A43 A47
rule_80: A42 -> 23 22
rule_81: A43 -> 1
rule_82: A43 -> A19 A44
rule_83: A44 -> 21 A34
rule_84: A45 -> A46 22
rule_85: A46 -> A21
rule_86: A47 -> 23
rule_87: A48 -> A49 14
rule_88: A49 -> A4 A13 A50
rule_89: A50 -> 7 8
rule_90: A51 -> 1
rule_91: A51 -> 1 A52 A56
rule_92: A51 -> A53 A52
rule_93: A52 -> 1
rule_94: A53 -> 1
rule_95: A54 -> 1
rule_96: A55 -> 1
rule_97: A56 -> A54
rule_98: A56 -> A55
rule_99: A57 -> 1
rule_100: A57 -> 1 A58 A65
rule_101: A57 -> A60 A58
rule_102: A57 -> A61 A58
rule_103: A57 -> A62 A58 24
rule_104: A57 -> A63 A58
rule_105: A57 -> A66 A58
rule_106: A57 -> A69 A58 24
rule_107: A58 -> 1
rule_108: A58 -> 1 27 A64
rule_109: A58 -> 1 A59
rule_110: A59 -> A61
rule_111: A59 -> A63
rule_112: A60 -> 1
rule_113: A60 -> 26
rule_114: A61 -> 1
rule_115: A62 -> 1
rule_116: A63 -> 1
rule_117: A64 -> A60 28
rule_118: A64 -> A62
rule_119: A65 -> A60
rule_120: A65 -> A62
rule_121: A65 -> A63
rule_122: A66 -> A67
rule_123: A67 -> A4 A10 A68
rule_124: A68 -> 5
rule_125: A69 -> A70
rule_126: A69 -> A71 26
rule_127: A70 -> A60 1
rule_128: A71 -> A5 A10 A72
rule_129: A72 -> 5
rule_130: A73 -> 1
rule_131: A73 -> 1 A74 A76
rule_132: A73 -> A20 A74 A75
rule_133: A73 -> A81 A74 A80
rule_134: A74 -> 1
rule_135: A75 -> 30
rule_136: A76 -> A77
rule_137: A76 -> A78 29
rule_138: A76 -> A79
rule_139: A77 -> A20
rule_140: A77 -> A21
rule_141: A77 -> A23
rule_142: A78 -> A20 30
rule_143: A78 -> A21
rule_144: A78 -> A23
rule_145: A79 -> A20
rule_146: A79 -> A22
rule_147: A80 -> 30 29
rule_148: A81 -> A82 13
rule_149: A82 -> A60 1 28
rule_150: A82 -> A60 1 28

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