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 42 terminals, ranging from 1 to 42.
Each terminal symbol occurs in the target sequence at least once.

There are 94 non-terminal symbols, ranging from A1 to A94.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 41 37 A81 A90 A43 40 A43 39 A89 38 37
rule_3: A2 -> 41 37 A81 A92 A43 A43 39 A91 38 37
rule_4: A2 -> 41 37 A81 A94 A43 40 A43 A93
rule_5: A2 -> 41 A81 A85 A43 A43 39 A84
rule_6: A3 -> A4 A5
rule_7: A4 -> 9
rule_8: A5 -> 7 A6
rule_9: A5 -> 8 A7
rule_10: A6 -> 1
rule_11: A6 -> 2
rule_12: A6 -> 3
rule_13: A7 -> 1
rule_14: A7 -> 2
rule_15: A7 -> 3
rule_16: A8 -> A9 A10
rule_17: A9 -> 10
rule_18: A10 -> 7 A11
rule_19: A10 -> 8 A12
rule_20: A11 -> 1
rule_21: A11 -> 2
rule_22: A11 -> 3
rule_23: A12 -> 1
rule_24: A12 -> 2
rule_25: A12 -> 3
rule_26: A13 -> A14 A15
rule_27: A14 -> 11
rule_28: A15 -> 4 A16
rule_29: A16 -> 1
rule_30: A16 -> 2
rule_31: A16 -> 3
rule_32: A17 -> 12
rule_33: A18 -> A19
rule_34: A18 -> A20
rule_35: A18 -> A21
rule_36: A18 -> A22
rule_37: A18 -> A23
rule_38: A18 -> A24
rule_39: A18 -> A25
rule_40: A18 -> A26
rule_41: A18 -> A27
rule_42: A18 -> A28
rule_43: A18 -> A29
rule_44: A18 -> A30
rule_45: A19 -> 5
rule_46: A20 -> 5
rule_47: A21 -> 5
rule_48: A22 -> 5
rule_49: A23 -> 5
rule_50: A24 -> 5
rule_51: A25 -> 5
rule_52: A26 -> 5
rule_53: A26 -> 13
rule_54: A27 -> 5
rule_55: A28 -> 5
rule_56: A29 -> 5
rule_57: A30 -> 5
rule_58: A31 -> 14 A32
rule_59: A32 -> 6
rule_60: A32 -> 15
rule_61: A33 -> A34 A35
rule_62: A34 -> 18
rule_63: A35 -> 16 A36
rule_64: A35 -> 17 A37
rule_65: A36 -> 1
rule_66: A36 -> 2
rule_67: A36 -> 3
rule_68: A36 -> 19
rule_69: A37 -> 1
rule_70: A37 -> 2
rule_71: A37 -> 3
rule_72: A38 -> A39 A40
rule_73: A39 -> 20
rule_74: A40 -> 16 A41
rule_75: A40 -> 17 A42
rule_76: A41 -> 1
rule_77: A41 -> 2
rule_78: A41 -> 3
rule_79: A42 -> 1
rule_80: A42 -> 2
rule_81: A42 -> 3
rule_82: A42 -> 21
rule_83: A43 -> 23
rule_84: A43 -> A44 A45
rule_85: A44 -> 24
rule_86: A45 -> 22 A46
rule_87: A46 -> 1
rule_88: A46 -> 2
rule_89: A46 -> 3
rule_90: A46 -> 25
rule_91: A47 -> A48 A31 A13 A18
rule_92: A48 -> 26 A17
rule_93: A49 -> A50
rule_94: A49 -> A51
rule_95: A49 -> A52
rule_96: A49 -> A53
rule_97: A49 -> A54
rule_98: A49 -> A55
rule_99: A49 -> A56
rule_100: A49 -> A57
rule_101: A49 -> A58
rule_102: A49 -> A59
rule_103: A49 -> A60
rule_104: A49 -> A61
rule_105: A50 -> 5
rule_106: A51 -> 5
rule_107: A52 -> 5
rule_108: A53 -> 5
rule_109: A54 -> 5
rule_110: A55 -> 5
rule_111: A56 -> 5
rule_112: A57 -> 5
rule_113: A58 -> 5
rule_114: A59 -> 5
rule_115: A60 -> 5
rule_116: A61 -> 5
rule_117: A62 -> 27 A63
rule_118: A63 -> 6
rule_119: A64 -> 28 A47 29
rule_120: A64 -> A65
rule_121: A64 -> A66
rule_122: A64 -> A67
rule_123: A64 -> A68
rule_124: A64 -> A69
rule_125: A64 -> A70
rule_126: A64 -> A71
rule_127: A64 -> A72
rule_128: A64 -> A73
rule_129: A64 -> A74
rule_130: A64 -> A75
rule_131: A64 -> A76
rule_132: A65 -> 5
rule_133: A66 -> 5
rule_134: A67 -> 5
rule_135: A68 -> 5
rule_136: A69 -> 5
rule_137: A70 -> 5
rule_138: A71 -> 5
rule_139: A72 -> 5
rule_140: A73 -> 5
rule_141: A74 -> 5
rule_142: A75 -> 5
rule_143: A76 -> 5
rule_144: A77 -> 30 A78
rule_145: A78 -> 6
rule_146: A78 -> 31
rule_147: A79 -> A80 A62 A3 A49
rule_148: A80 -> 32 A8
rule_149: A81 -> A82 A77 A38 A64
rule_150: A82 -> 33 A33
rule_151: A83 -> 34
rule_152: A84 -> A81
rule_153: A84 -> A83 35
rule_154: A85 -> A79 A86
rule_155: A85 -> A81 A87
rule_156: A85 -> A83 A88
rule_157: A86 -> 36
rule_158: A87 -> 36
rule_159: A88 -> 36
rule_160: A89 -> A79
rule_161: A89 -> A83
rule_162: A90 -> A79 A86
rule_163: A90 -> A81 A87
rule_164: A90 -> A83 A88
rule_165: A91 -> A83
rule_166: A92 -> A79 A86
rule_167: A92 -> A81 A87
rule_168: A92 -> A83 A88
rule_169: A93 -> A83 35
rule_170: A94 -> A79 A86
rule_171: A94 -> A81 A87
rule_172: A94 -> A83 A88

Target sequence:
41 37 33 18 16 19 30 31 20 17 21 28 26 12 14 15 11 4 1 13 29 32 10 8 2 27 6 9 7 3 5 36 24 22 25 40 42 23 39 34 35 38 37
"""