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 43 terminals, ranging from 1 to 43.
Each terminal symbol occurs in the target sequence at least once.

There are 86 non-terminal symbols, ranging from A1 to A86.

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

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