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 89 non-terminal symbols, ranging from A1 to A89.

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

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