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 35 terminals, ranging from 1 to 35.
Each terminal symbol occurs in the target sequence at least once.

There are 64 non-terminal symbols, ranging from A1 to A64.

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

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