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 45 terminals, ranging from 1 to 45.
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 -> 44 40 A86 A98 A48 A48 A97 41 40
rule_3: A2 -> 44 40 A86 A100 A48 A48 42 A99
rule_4: A2 -> 44 A86 A94 A48 43 45 A48 42 A93 40
rule_5: A2 -> 44 A86 A96 A48 43 A48 42 A95 41
rule_6: A3 -> A4 A5
rule_7: A4 -> 11
rule_8: A5 -> 9 A6
rule_9: A5 -> 10 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 -> 12
rule_18: A10 -> 9 A11
rule_19: A10 -> 10 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 -> 13
rule_28: A14 -> 14
rule_29: A15 -> 4 A16
rule_30: A15 -> 5 A17
rule_31: A15 -> 6 A18
rule_32: A16 -> 1
rule_33: A16 -> 2
rule_34: A16 -> 3
rule_35: A16 -> 15
rule_36: A17 -> 1
rule_37: A17 -> 2
rule_38: A17 -> 3
rule_39: A18 -> 1
rule_40: A18 -> 2
rule_41: A18 -> 3
rule_42: A19 -> A20 A21
rule_43: A20 -> 16
rule_44: A21 -> 4 A22
rule_45: A21 -> 5 A23
rule_46: A21 -> 6 A24
rule_47: A22 -> 1
rule_48: A22 -> 2
rule_49: A22 -> 3
rule_50: A23 -> 1
rule_51: A23 -> 2
rule_52: A23 -> 3
rule_53: A23 -> 17
rule_54: A24 -> 1
rule_55: A24 -> 2
rule_56: A24 -> 3
rule_57: A25 -> A26 A27
rule_58: A26 -> 18
rule_59: A27 -> 4 A28
rule_60: A27 -> 5 A29
rule_61: A27 -> 6 A30
rule_62: A28 -> 1
rule_63: A28 -> 2
rule_64: A28 -> 3
rule_65: A29 -> 1
rule_66: A29 -> 2
rule_67: A29 -> 3
rule_68: A30 -> 1
rule_69: A30 -> 2
rule_70: A30 -> 3
rule_71: A30 -> 19
rule_72: A31 -> A32
rule_73: A31 -> A33
rule_74: A31 -> A34
rule_75: A31 -> A35
rule_76: A31 -> A36
rule_77: A31 -> A37
rule_78: A31 -> A38
rule_79: A31 -> A39
rule_80: A31 -> A40
rule_81: A31 -> A41
rule_82: A31 -> A42
rule_83: A31 -> A43
rule_84: A32 -> 7
rule_85: A33 -> 7
rule_86: A34 -> 7
rule_87: A35 -> 7
rule_88: A36 -> 7
rule_89: A37 -> 7
rule_90: A38 -> 7
rule_91: A39 -> 7
rule_92: A39 -> 20
rule_93: A40 -> 7
rule_94: A41 -> 7
rule_95: A42 -> 7
rule_96: A43 -> 7
rule_97: A44 -> 21 A45
rule_98: A45 -> 8
rule_99: A45 -> 22
rule_100: A46 -> 23 A47
rule_101: A47 -> 1
rule_102: A47 -> 2
rule_103: A47 -> 3
rule_104: A47 -> 24
rule_105: A48 -> 26
rule_106: A48 -> A49 A50
rule_107: A49 -> 27
rule_108: A50 -> 25 A51
rule_109: A51 -> 1
rule_110: A51 -> 2
rule_111: A51 -> 3
rule_112: A51 -> 28
rule_113: A52 -> A53 A44 A13 A31
rule_114: A53 -> 29 A25
rule_115: A54 -> A55
rule_116: A54 -> A56
rule_117: A54 -> A57
rule_118: A54 -> A58
rule_119: A54 -> A59
rule_120: A54 -> A60
rule_121: A54 -> A61
rule_122: A54 -> A62
rule_123: A54 -> A63
rule_124: A54 -> A64
rule_125: A54 -> A65
rule_126: A54 -> A66
rule_127: A55 -> 7
rule_128: A56 -> 7
rule_129: A57 -> 7
rule_130: A58 -> 7
rule_131: A59 -> 7
rule_132: A60 -> 7
rule_133: A61 -> 7
rule_134: A62 -> 7
rule_135: A63 -> 7
rule_136: A64 -> 7
rule_137: A65 -> 7
rule_138: A66 -> 7
rule_139: A67 -> 30 A68
rule_140: A68 -> 8
rule_141: A69 -> 31 A52 32
rule_142: A69 -> A70
rule_143: A69 -> A71
rule_144: A69 -> A72
rule_145: A69 -> A73
rule_146: A69 -> A74
rule_147: A69 -> A75
rule_148: A69 -> A76
rule_149: A69 -> A77
rule_150: A69 -> A78
rule_151: A69 -> A79
rule_152: A69 -> A80
rule_153: A69 -> A81
rule_154: A70 -> 7
rule_155: A71 -> 7
rule_156: A72 -> 7
rule_157: A73 -> 7
rule_158: A74 -> 7
rule_159: A75 -> 7
rule_160: A76 -> 7
rule_161: A77 -> 7
rule_162: A78 -> 7
rule_163: A79 -> 7
rule_164: A80 -> 7
rule_165: A81 -> 7
rule_166: A82 -> 33 A83
rule_167: A83 -> 8
rule_168: A83 -> 34
rule_169: A84 -> A85 A67 A3 A54
rule_170: A85 -> 35 A8
rule_171: A86 -> A87 A82 A13 A69
rule_172: A87 -> 36 A19
rule_173: A88 -> A89 A46
rule_174: A89 -> 37
rule_175: A90 -> 39
rule_176: A91 -> 39
rule_177: A92 -> 39
rule_178: A93 -> A88
rule_179: A94 -> A84 A90
rule_180: A94 -> A86 A91
rule_181: A94 -> A88 A92
rule_182: A95 -> A88
rule_183: A96 -> A84 A90
rule_184: A96 -> A86 A91
rule_185: A96 -> A88 A92
rule_186: A97 -> A86
rule_187: A97 -> A88 38
rule_188: A98 -> A84 A90
rule_189: A98 -> A86 A91
rule_190: A98 -> A88 A92
rule_191: A99 -> A88 38
rule_192: A100 -> A84 A90
rule_193: A100 -> A86 A91
rule_194: A100 -> A88 A92

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