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 46 terminals, ranging from 1 to 46.
Each terminal symbol occurs in the target sequence at least once.

There are 100 non-terminal symbols, ranging from A1 to A100.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 46 A100 A89 39 A90 A99
rule_3: A3 -> A4 A5
rule_4: A4 -> 13
rule_5: A5 -> 11 A6
rule_6: A5 -> 12 A7
rule_7: A6 -> 1
rule_8: A6 -> 2
rule_9: A6 -> 3
rule_10: A6 -> 4
rule_11: A6 -> 5
rule_12: A7 -> 1
rule_13: A7 -> 2
rule_14: A7 -> 3
rule_15: A7 -> 4
rule_16: A7 -> 5
rule_17: A8 -> A9 A10
rule_18: A9 -> 14
rule_19: A10 -> 11 A11
rule_20: A10 -> 12 A12
rule_21: A11 -> 1
rule_22: A11 -> 2
rule_23: A11 -> 3
rule_24: A11 -> 4
rule_25: A11 -> 5
rule_26: A11 -> 15
rule_27: A12 -> 1
rule_28: A12 -> 2
rule_29: A12 -> 3
rule_30: A12 -> 4
rule_31: A12 -> 5
rule_32: A13 -> A14 A15
rule_33: A14 -> 16
rule_34: A15 -> 6 A16
rule_35: A15 -> 7 A17
rule_36: A16 -> 1
rule_37: A16 -> 2
rule_38: A16 -> 3
rule_39: A16 -> 4
rule_40: A16 -> 5
rule_41: A17 -> 1
rule_42: A17 -> 2
rule_43: A17 -> 3
rule_44: A17 -> 4
rule_45: A17 -> 5
rule_46: A18 -> A19 A20
rule_47: A19 -> 17
rule_48: A20 -> 6 A21
rule_49: A20 -> 7 A22
rule_50: A21 -> 1
rule_51: A21 -> 2
rule_52: A21 -> 3
rule_53: A21 -> 4
rule_54: A21 -> 5
rule_55: A22 -> 1
rule_56: A22 -> 2
rule_57: A22 -> 3
rule_58: A22 -> 4
rule_59: A22 -> 5
rule_60: A23 -> 18 A24
rule_61: A24 -> 1
rule_62: A24 -> 2
rule_63: A24 -> 3
rule_64: A24 -> 4
rule_65: A24 -> 5
rule_66: A25 -> A26 A27
rule_67: A26 -> 21
rule_68: A27 -> 19 A28
rule_69: A27 -> 20 A29
rule_70: A28 -> 1
rule_71: A28 -> 2
rule_72: A28 -> 3
rule_73: A28 -> 4
rule_74: A28 -> 5
rule_75: A29 -> 1
rule_76: A29 -> 2
rule_77: A29 -> 3
rule_78: A29 -> 4
rule_79: A29 -> 5
rule_80: A30 -> 22
rule_81: A31 -> 23
rule_82: A31 -> A32 A33
rule_83: A32 -> 24
rule_84: A33 -> 19 A34
rule_85: A33 -> 20 A35
rule_86: A34 -> 1
rule_87: A34 -> 2
rule_88: A34 -> 3
rule_89: A34 -> 4
rule_90: A34 -> 5
rule_91: A35 -> 1
rule_92: A35 -> 2
rule_93: A35 -> 3
rule_94: A35 -> 4
rule_95: A35 -> 5
rule_96: A35 -> 25
rule_97: A36 -> A37
rule_98: A36 -> A38
rule_99: A36 -> A39
rule_100: A36 -> A40
rule_101: A36 -> A41
rule_102: A36 -> A42
rule_103: A36 -> A43
rule_104: A36 -> A44
rule_105: A36 -> A45
rule_106: A36 -> A46
rule_107: A36 -> A47
rule_108: A36 -> A48
rule_109: A37 -> 8
rule_110: A38 -> 8
rule_111: A39 -> 8
rule_112: A40 -> 8
rule_113: A40 -> 27
rule_114: A41 -> 8
rule_115: A42 -> 8
rule_116: A43 -> 8
rule_117: A44 -> 8
rule_118: A45 -> 8
rule_119: A46 -> 8
rule_120: A47 -> 8
rule_121: A48 -> 8
rule_122: A49 -> 28 A50
rule_123: A50 -> 9
rule_124: A50 -> 10
rule_125: A51 -> A52
rule_126: A51 -> A53
rule_127: A51 -> A54
rule_128: A51 -> A55
rule_129: A51 -> A56
rule_130: A51 -> A57
rule_131: A51 -> A58
rule_132: A51 -> A59
rule_133: A51 -> A60
rule_134: A51 -> A61
rule_135: A51 -> A62
rule_136: A51 -> A63
rule_137: A52 -> 8
rule_138: A53 -> 8
rule_139: A54 -> 8
rule_140: A55 -> 8
rule_141: A56 -> 8
rule_142: A57 -> 8
rule_143: A58 -> 8
rule_144: A58 -> 29
rule_145: A59 -> 8
rule_146: A60 -> 8
rule_147: A61 -> 8
rule_148: A62 -> 8
rule_149: A63 -> 8
rule_150: A64 -> 30 A65
rule_151: A65 -> 9
rule_152: A65 -> 10
rule_153: A65 -> 31
rule_154: A66 -> A67
rule_155: A66 -> A68
rule_156: A66 -> A69
rule_157: A66 -> A70
rule_158: A66 -> A71
rule_159: A66 -> A72
rule_160: A66 -> A73
rule_161: A66 -> A74
rule_162: A66 -> A75
rule_163: A66 -> A76
rule_164: A66 -> A77
rule_165: A66 -> A78
rule_166: A67 -> 8
rule_167: A68 -> 8
rule_168: A69 -> 8
rule_169: A70 -> 8
rule_170: A71 -> 8
rule_171: A72 -> 8
rule_172: A73 -> 8
rule_173: A74 -> 8
rule_174: A75 -> 8
rule_175: A76 -> 8
rule_176: A77 -> 8
rule_177: A78 -> 8
rule_178: A79 -> 32 A80
rule_179: A80 -> 9
rule_180: A80 -> 10
rule_181: A81 -> A82 A23
rule_182: A82 -> 33
rule_183: A83 -> A84 A79 A18 A66
rule_184: A84 -> 34 A13
rule_185: A85 -> A86 A49 A25 A36
rule_186: A86 -> 35 A31
rule_187: A87 -> 36
rule_188: A87 -> A88 A64 A31 A51
rule_189: A88 -> 37 A30
rule_190: A89 -> 38 A83
rule_191: A90 -> A94
rule_192: A91 -> 40 A85
rule_193: A92 -> 42 A87 A93
rule_194: A93 -> 26
rule_195: A94 -> A95 A92 A3 A91
rule_196: A95 -> 43 A8
rule_197: A96 -> 45
rule_198: A97 -> 45
rule_199: A98 -> 45
rule_200: A99 -> A83
rule_201: A99 -> A87
rule_202: A100 -> A81 A97
rule_203: A100 -> A83 A96
rule_204: A100 -> A87 A98

Target sequence:
46 33 18 1 45 38 34 16 6 2 32 9 17 7 3 8 39 43 14 11 15 42 37 22 30 31 24 20 25 29 26 13 12 4 40 35 23 28 10 21 19 5 27 41 36 44
"""