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 41 terminals, ranging from 1 to 41.
Each terminal symbol occurs in the target sequence at least once.

There are 97 non-terminal symbols, ranging from A1 to A97.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 41 A92 A95
rule_3: A3 -> A4 A5
rule_4: A4 -> 7
rule_5: A5 -> 5 A6
rule_6: A5 -> 6 A7
rule_7: A6 -> 1
rule_8: A6 -> 2
rule_9: A6 -> 3
rule_10: A6 -> 4
rule_11: A7 -> 1
rule_12: A7 -> 2
rule_13: A7 -> 3
rule_14: A7 -> 4
rule_15: A8 -> A9 A10
rule_16: A9 -> 8
rule_17: A10 -> 5 A11
rule_18: A10 -> 6 A12
rule_19: A11 -> 1
rule_20: A11 -> 2
rule_21: A11 -> 3
rule_22: A11 -> 4
rule_23: A12 -> 1
rule_24: A12 -> 2
rule_25: A12 -> 3
rule_26: A12 -> 4
rule_27: A13 -> A14 A15
rule_28: A14 -> 13
rule_29: A15 -> 11 A16
rule_30: A15 -> 12 A17
rule_31: A16 -> 1
rule_32: A16 -> 2
rule_33: A16 -> 3
rule_34: A16 -> 4
rule_35: A17 -> 1
rule_36: A17 -> 2
rule_37: A17 -> 3
rule_38: A17 -> 4
rule_39: A17 -> 14
rule_40: A18 -> A19 A20
rule_41: A19 -> 15
rule_42: A20 -> 11 A21
rule_43: A20 -> 12 A22
rule_44: A21 -> 1
rule_45: A21 -> 2
rule_46: A21 -> 3
rule_47: A21 -> 4
rule_48: A22 -> 1
rule_49: A22 -> 2
rule_50: A22 -> 3
rule_51: A22 -> 4
rule_52: A23 -> A24 A25
rule_53: A24 -> 19
rule_54: A25 -> 16 A26
rule_55: A25 -> 17 A27
rule_56: A25 -> 18 A28
rule_57: A26 -> 1
rule_58: A26 -> 2
rule_59: A26 -> 3
rule_60: A26 -> 4
rule_61: A26 -> 20
rule_62: A27 -> 1
rule_63: A27 -> 2
rule_64: A27 -> 3
rule_65: A27 -> 4
rule_66: A28 -> 1
rule_67: A28 -> 2
rule_68: A28 -> 3
rule_69: A28 -> 4
rule_70: A29 -> A30 A31
rule_71: A30 -> 21
rule_72: A30 -> 22
rule_73: A31 -> 16 A32
rule_74: A31 -> 17 A33
rule_75: A31 -> 18 A34
rule_76: A32 -> 1
rule_77: A32 -> 2
rule_78: A32 -> 3
rule_79: A32 -> 4
rule_80: A33 -> 1
rule_81: A33 -> 2
rule_82: A33 -> 3
rule_83: A33 -> 4
rule_84: A34 -> 1
rule_85: A34 -> 2
rule_86: A34 -> 3
rule_87: A34 -> 4
rule_88: A34 -> 23
rule_89: A35 -> A36 A37
rule_90: A36 -> 24
rule_91: A37 -> 16 A38
rule_92: A37 -> 17 A39
rule_93: A37 -> 18 A40
rule_94: A38 -> 1
rule_95: A38 -> 2
rule_96: A38 -> 3
rule_97: A38 -> 4
rule_98: A38 -> 25
rule_99: A39 -> 1
rule_100: A39 -> 2
rule_101: A39 -> 3
rule_102: A39 -> 4
rule_103: A40 -> 1
rule_104: A40 -> 2
rule_105: A40 -> 3
rule_106: A40 -> 4
rule_107: A41 -> A42
rule_108: A41 -> A43
rule_109: A41 -> A44
rule_110: A41 -> A45
rule_111: A41 -> A46
rule_112: A41 -> A47
rule_113: A41 -> A48
rule_114: A41 -> A49
rule_115: A41 -> A50
rule_116: A41 -> A51
rule_117: A41 -> A52
rule_118: A41 -> A53
rule_119: A42 -> 9
rule_120: A43 -> 9
rule_121: A44 -> 9
rule_122: A45 -> 9
rule_123: A46 -> 9
rule_124: A47 -> 9
rule_125: A48 -> 9
rule_126: A49 -> 9
rule_127: A50 -> 9
rule_128: A51 -> 9
rule_129: A52 -> 9
rule_130: A53 -> 9
rule_131: A54 -> 26 A55
rule_132: A55 -> 10
rule_133: A56 -> A57
rule_134: A56 -> A58
rule_135: A56 -> A59
rule_136: A56 -> A60
rule_137: A56 -> A61
rule_138: A56 -> A62
rule_139: A56 -> A63
rule_140: A56 -> A64
rule_141: A56 -> A65
rule_142: A56 -> A66
rule_143: A56 -> A67
rule_144: A56 -> A68
rule_145: A57 -> 9
rule_146: A58 -> 9
rule_147: A59 -> 9
rule_148: A60 -> 9
rule_149: A61 -> 9
rule_150: A62 -> 9
rule_151: A63 -> 9
rule_152: A64 -> 9
rule_153: A65 -> 9
rule_154: A65 -> 27
rule_155: A66 -> 9
rule_156: A67 -> 9
rule_157: A68 -> 9
rule_158: A69 -> 28 A70
rule_159: A70 -> 10
rule_160: A70 -> 29
rule_161: A71 -> A72 A69 A29 A56
rule_162: A72 -> 30 A35
rule_163: A73 -> A74
rule_164: A73 -> A75
rule_165: A73 -> A76
rule_166: A73 -> A77
rule_167: A73 -> A78
rule_168: A73 -> A79
rule_169: A73 -> A80
rule_170: A73 -> A81
rule_171: A73 -> A82
rule_172: A73 -> A83
rule_173: A73 -> A84
rule_174: A73 -> A85
rule_175: A74 -> 9
rule_176: A75 -> 9
rule_177: A76 -> 9
rule_178: A77 -> 9
rule_179: A78 -> 9
rule_180: A79 -> 9
rule_181: A80 -> 9
rule_182: A80 -> 31
rule_183: A81 -> 9
rule_184: A82 -> 9
rule_185: A83 -> 9
rule_186: A84 -> 9
rule_187: A85 -> 9
rule_188: A86 -> 32 A87
rule_189: A87 -> 10
rule_190: A87 -> 33
rule_191: A88 -> A89 A86 A13 A73
rule_192: A89 -> 34 A18
rule_193: A90 -> 35 A88 36
rule_194: A91 -> 37 A71
rule_195: A92 -> A93
rule_196: A93 -> A94 A54 A3 A41
rule_197: A94 -> 39 A8
rule_198: A95 -> A96
rule_199: A96 -> A97 A91 A23 A90
rule_200: A97 -> 40 A29

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