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 39 terminals, ranging from 1 to 39.
Each terminal symbol occurs in the target sequence at least once.

There are 67 non-terminal symbols, ranging from A1 to A67.

Production rules:
rule_1: A1 -> A64 A60 A55 A53 A51 A39 A36 A28 A25 A15 A3 A2
rule_2: A2 -> 1
rule_3: A2 -> 2
rule_4: A3 -> A5 A4
rule_5: A3 -> A7 A6
rule_6: A3 -> A9 A8 3
rule_7: A4 -> 2
rule_8: A5 -> 2
rule_9: A6 -> 2
rule_10: A7 -> 2
rule_11: A8 -> 2
rule_12: A8 -> 4
rule_13: A9 -> 2
rule_14: A9 -> 7 A14
rule_15: A9 -> A13 A10
rule_16: A10 -> 2
rule_17: A10 -> 2
rule_18: A10 -> 2
rule_19: A10 -> 2
rule_20: A10 -> 2
rule_21: A10 -> 2
rule_22: A10 -> 2
rule_23: A10 -> 2
rule_24: A10 -> 2
rule_25: A10 -> 2
rule_26: A10 -> 2
rule_27: A10 -> 2
rule_28: A10 -> 2
rule_29: A10 -> 2
rule_30: A10 -> 2
rule_31: A10 -> 2
rule_32: A10 -> A11
rule_33: A11 -> 2
rule_34: A11 -> A9 A12 5
rule_35: A12 -> 2
rule_36: A12 -> 6
rule_37: A13 -> 8
rule_38: A14 -> 2
rule_39: A14 -> 2
rule_40: A14 -> 2
rule_41: A14 -> 2
rule_42: A14 -> 2
rule_43: A14 -> 2
rule_44: A14 -> 2
rule_45: A14 -> 2
rule_46: A14 -> 2
rule_47: A14 -> 2
rule_48: A14 -> 2
rule_49: A14 -> 2
rule_50: A14 -> 2
rule_51: A14 -> 2
rule_52: A14 -> 2
rule_53: A14 -> 2
rule_54: A14 -> A11
rule_55: A15 -> A17 A16
rule_56: A15 -> A19 A18
rule_57: A16 -> 2
rule_58: A17 -> 2
rule_59: A18 -> 2
rule_60: A18 -> 10
rule_61: A19 -> 2
rule_62: A19 -> 13 A22
rule_63: A20 -> 2
rule_64: A20 -> A19 A21 11
rule_65: A21 -> 2
rule_66: A21 -> 12
rule_67: A22 -> 2
rule_68: A22 -> 2
rule_69: A22 -> 2
rule_70: A22 -> 2
rule_71: A22 -> 2
rule_72: A22 -> 2
rule_73: A22 -> 2
rule_74: A22 -> 2
rule_75: A22 -> 2
rule_76: A22 -> 2
rule_77: A22 -> 2
rule_78: A22 -> 2
rule_79: A22 -> 2
rule_80: A22 -> 2
rule_81: A22 -> 2
rule_82: A22 -> 2
rule_83: A22 -> A20
rule_84: A23 -> 2
rule_85: A24 -> 2
rule_86: A25 -> A17 A26
rule_87: A25 -> A19 A27
rule_88: A26 -> 2
rule_89: A27 -> 2
rule_90: A27 -> 15
rule_91: A28 -> A9 A35
rule_92: A28 -> A19 A34
rule_93: A28 -> A30 A29
rule_94: A29 -> 2
rule_95: A29 -> 17
rule_96: A30 -> 2
rule_97: A30 -> 20 A33
rule_98: A31 -> 2
rule_99: A31 -> A30 A32 18
rule_100: A32 -> 2
rule_101: A32 -> 19
rule_102: A33 -> 2
rule_103: A33 -> 2
rule_104: A33 -> 2
rule_105: A33 -> 2
rule_106: A33 -> 2
rule_107: A33 -> 2
rule_108: A33 -> 2
rule_109: A33 -> 2
rule_110: A33 -> 2
rule_111: A33 -> 2
rule_112: A33 -> 2
rule_113: A33 -> 2
rule_114: A33 -> 2
rule_115: A33 -> 2
rule_116: A33 -> 2
rule_117: A33 -> 2
rule_118: A33 -> A31
rule_119: A34 -> 2
rule_120: A35 -> 2
rule_121: A36 -> A23 A38
rule_122: A36 -> A30 A37 21
rule_123: A37 -> 2
rule_124: A37 -> 22
rule_125: A38 -> 2
rule_126: A39 -> A41 A40
rule_127: A40 -> 2
rule_128: A40 -> 24
rule_129: A41 -> 2
rule_130: A41 -> 26 A44
rule_131: A42 -> 2
rule_132: A42 -> A41 A43 25
rule_133: A43 -> 2
rule_134: A44 -> 2
rule_135: A44 -> 2
rule_136: A44 -> 2
rule_137: A44 -> 2
rule_138: A44 -> 2
rule_139: A44 -> 2
rule_140: A44 -> 2
rule_141: A44 -> 2
rule_142: A44 -> 2
rule_143: A44 -> 2
rule_144: A44 -> 2
rule_145: A44 -> 2
rule_146: A44 -> 2
rule_147: A44 -> 2
rule_148: A44 -> 2
rule_149: A44 -> 2
rule_150: A44 -> A42
rule_151: A45 -> 2
rule_152: A46 -> 2
rule_153: A46 -> A13 29 A47
rule_154: A46 -> 29 A50
rule_155: A47 -> 2
rule_156: A47 -> 2
rule_157: A47 -> 2
rule_158: A47 -> 2
rule_159: A47 -> 2
rule_160: A47 -> 2
rule_161: A47 -> 2
rule_162: A47 -> 2
rule_163: A47 -> 2
rule_164: A47 -> 2
rule_165: A47 -> 2
rule_166: A47 -> 2
rule_167: A47 -> 2
rule_168: A47 -> 2
rule_169: A47 -> 2
rule_170: A47 -> 2
rule_171: A47 -> A48
rule_172: A48 -> 2
rule_173: A48 -> A46 A49 27
rule_174: A49 -> 2
rule_175: A49 -> 28
rule_176: A50 -> 2
rule_177: A50 -> 2
rule_178: A50 -> 2
rule_179: A50 -> 2
rule_180: A50 -> 2
rule_181: A50 -> 2
rule_182: A50 -> 2
rule_183: A50 -> 2
rule_184: A50 -> 2
rule_185: A50 -> 2
rule_186: A50 -> 2
rule_187: A50 -> 2
rule_188: A50 -> 2
rule_189: A50 -> 2
rule_190: A50 -> 2
rule_191: A50 -> 2
rule_192: A50 -> A48
rule_193: A51 -> A41 A52 31
rule_194: A52 -> 2
rule_195: A52 -> 30
rule_196: A53 -> A30 A54
rule_197: A54 -> 2
rule_198: A54 -> 33
rule_199: A55 -> A19 A57
rule_200: A55 -> A23 A58
rule_201: A55 -> A24 A59
rule_202: A55 -> A30 A56 34
rule_203: A56 -> 2
rule_204: A56 -> 35
rule_205: A57 -> 2
rule_206: A58 -> 2
rule_207: A59 -> 2
rule_208: A60 -> A23 A62
rule_209: A60 -> A41 A61
rule_210: A60 -> A45 A62
rule_211: A60 -> A46 A63
rule_212: A61 -> 2
rule_213: A62 -> 2
rule_214: A63 -> 2
rule_215: A63 -> 37
rule_216: A64 -> A19 A66
rule_217: A64 -> A23 A67
rule_218: A64 -> A30 A65 38
rule_219: A65 -> 2
rule_220: A65 -> 39
rule_221: A66 -> 2
rule_222: A67 -> 2

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