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 42 terminals, ranging from 1 to 42.
Each terminal symbol occurs in the target sequence at least once.

There are 83 non-terminal symbols, ranging from A1 to A83.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 42 A83 A72 A73 A74 A82
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 -> A24 A25
rule_61: A24 -> 20
rule_62: A25 -> 17 A26
rule_63: A25 -> 18 A28
rule_64: A25 -> 19 A27
rule_65: A26 -> 1
rule_66: A26 -> 2
rule_67: A26 -> 3
rule_68: A26 -> 4
rule_69: A26 -> 5
rule_70: A27 -> 1
rule_71: A27 -> 2
rule_72: A27 -> 3
rule_73: A27 -> 4
rule_74: A27 -> 5
rule_75: A27 -> 21
rule_76: A28 -> 1
rule_77: A28 -> 2
rule_78: A28 -> 3
rule_79: A28 -> 4
rule_80: A28 -> 5
rule_81: A29 -> A30 A31
rule_82: A30 -> 22
rule_83: A31 -> 17 A32
rule_84: A31 -> 18 A34
rule_85: A31 -> 19 A33
rule_86: A32 -> 1
rule_87: A32 -> 2
rule_88: A32 -> 3
rule_89: A32 -> 4
rule_90: A32 -> 5
rule_91: A33 -> 1
rule_92: A33 -> 2
rule_93: A33 -> 3
rule_94: A33 -> 4
rule_95: A33 -> 5
rule_96: A34 -> 1
rule_97: A34 -> 2
rule_98: A34 -> 3
rule_99: A34 -> 4
rule_100: A34 -> 5
rule_101: A35 -> 23
rule_102: A36 -> A37
rule_103: A36 -> A38
rule_104: A36 -> A39
rule_105: A36 -> A40
rule_106: A36 -> A41
rule_107: A36 -> A42
rule_108: A36 -> A43
rule_109: A36 -> A44
rule_110: A36 -> A45
rule_111: A36 -> A46
rule_112: A36 -> A47
rule_113: A36 -> A48
rule_114: A37 -> 10
rule_115: A38 -> 10
rule_116: A39 -> 10
rule_117: A40 -> 10
rule_118: A41 -> 10
rule_119: A42 -> 10
rule_120: A43 -> 10
rule_121: A44 -> 10
rule_122: A45 -> 10
rule_123: A46 -> 10
rule_124: A47 -> 10
rule_125: A48 -> 10
rule_126: A49 -> 24 A50
rule_127: A50 -> 11
rule_128: A51 -> A52
rule_129: A51 -> A53
rule_130: A51 -> A54
rule_131: A51 -> A55
rule_132: A51 -> A56
rule_133: A51 -> A57
rule_134: A51 -> A58
rule_135: A51 -> A59
rule_136: A51 -> A60
rule_137: A51 -> A61
rule_138: A51 -> A62
rule_139: A51 -> A63
rule_140: A52 -> 10
rule_141: A53 -> 10
rule_142: A54 -> 10
rule_143: A55 -> 10
rule_144: A56 -> 10
rule_145: A57 -> 10
rule_146: A58 -> 10
rule_147: A59 -> 10
rule_148: A60 -> 10
rule_149: A61 -> 10
rule_150: A62 -> 10
rule_151: A62 -> 25
rule_152: A63 -> 10
rule_153: A64 -> 26 A35 27
rule_154: A65 -> 28
rule_155: A65 -> A66 A25
rule_156: A66 -> 29
rule_157: A67 -> A68 A64 A23 A51
rule_158: A68 -> 30 A29
rule_159: A69 -> 31
rule_160: A70 -> A71 A49 A8 A36
rule_161: A71 -> 32 A3
rule_162: A72 -> 33 A70
rule_163: A73 -> 34
rule_164: A74 -> A75
rule_165: A75 -> A76 A78 A13 A77
rule_166: A76 -> 35 A18
rule_167: A77 -> 36 A69 37
rule_168: A78 -> 38 A67
rule_169: A79 -> 41
rule_170: A80 -> 41
rule_171: A81 -> 41
rule_172: A82 -> A65
rule_173: A82 -> A67
rule_174: A82 -> A70
rule_175: A83 -> A65 A80
rule_176: A83 -> A67 A81
rule_177: A83 -> A70 A79

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