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 29 terminals, ranging from 1 to 29.
Each terminal symbol occurs in the target sequence at least once.

There are 126 non-terminal symbols, ranging from A1 to A126.

Production rules:
rule_1: A1 -> A123 A121 A113 A108 A74 A2
rule_2: A2 -> A29 A3 A30
rule_3: A2 -> A67 A36 A68
rule_4: A3 -> A9
rule_5: A4 -> 1
rule_6: A4 -> 3 A5
rule_7: A4 -> 5 A6
rule_8: A4 -> 7 A7 6
rule_9: A4 -> 9 A8
rule_10: A5 -> 1
rule_11: A6 -> 1
rule_12: A7 -> 1
rule_13: A8 -> 1
rule_14: A9 -> 1
rule_15: A9 -> 3 A11 2
rule_16: A9 -> 5 A12
rule_17: A9 -> 7 A10
rule_18: A9 -> 9 A13 8
rule_19: A10 -> 1
rule_20: A11 -> 1
rule_21: A12 -> 1
rule_22: A13 -> 1
rule_23: A14 -> 1
rule_24: A14 -> 3 A15 2
rule_25: A14 -> 5 A16 4
rule_26: A14 -> 7 A17
rule_27: A14 -> 9 A18
rule_28: A15 -> 1
rule_29: A16 -> 1
rule_30: A17 -> 1
rule_31: A18 -> 1
rule_32: A19 -> 1
rule_33: A19 -> 3 A21
rule_34: A19 -> 5 A22 4
rule_35: A19 -> 7 A20 6
rule_36: A19 -> 9 A23 8
rule_37: A20 -> 1
rule_38: A21 -> 1
rule_39: A22 -> 1
rule_40: A23 -> 1
rule_41: A24 -> 1
rule_42: A24 -> 3 A25 2
rule_43: A24 -> 5 A26
rule_44: A24 -> 7 A27
rule_45: A24 -> 9 A28
rule_46: A25 -> 1
rule_47: A26 -> 1
rule_48: A27 -> 1
rule_49: A28 -> 1
rule_50: A29 -> A14
rule_51: A30 -> A19
rule_52: A30 -> A24
rule_53: A31 -> 1
rule_54: A31 -> 3 A33 2
rule_55: A31 -> 5 A34 4
rule_56: A31 -> 7 A32 6
rule_57: A31 -> 9 A35 8
rule_58: A32 -> 1
rule_59: A33 -> 1
rule_60: A34 -> 1
rule_61: A35 -> 1
rule_62: A36 -> A37 10
rule_63: A36 -> A42
rule_64: A37 -> 1
rule_65: A37 -> 3 A39
rule_66: A37 -> 5 A38 4
rule_67: A37 -> 7 A41
rule_68: A37 -> 9 A40
rule_69: A38 -> 1
rule_70: A38 -> 11
rule_71: A39 -> 1
rule_72: A39 -> 12 A38 4
rule_73: A39 -> 16 9 A40 8
rule_74: A40 -> 1
rule_75: A40 -> 13
rule_76: A40 -> 13 A39 2
rule_77: A40 -> 15 7 A41 6
rule_78: A41 -> 1
rule_79: A41 -> 14 9 A40 8
rule_80: A42 -> 1
rule_81: A42 -> 3 A44 2
rule_82: A42 -> 5 A45 4
rule_83: A42 -> 7 A46
rule_84: A42 -> 9 A43
rule_85: A43 -> 1
rule_86: A43 -> 13 A44
rule_87: A43 -> 15
rule_88: A43 -> 15 A46
rule_89: A44 -> 1
rule_90: A44 -> 12 5 A45 4
rule_91: A44 -> 16 9 A43
rule_92: A45 -> 1
rule_93: A46 -> 1
rule_94: A46 -> 14 A43 8
rule_95: A47 -> 1
rule_96: A47 -> 3 A49
rule_97: A47 -> 5 A50 4
rule_98: A47 -> 7 A51 6
rule_99: A47 -> 9 A48
rule_100: A48 -> 1
rule_101: A48 -> 13 3 A49 2
rule_102: A48 -> 15 7 A51 6
rule_103: A49 -> 1
rule_104: A49 -> 12 5 A50
rule_105: A49 -> 16 9 A48 8
rule_106: A50 -> 1
rule_107: A51 -> 1
rule_108: A51 -> 14 9 A48
rule_109: A52 -> 1
rule_110: A52 -> 3 A54
rule_111: A52 -> 5 A55 4
rule_112: A52 -> 7 A56 6
rule_113: A52 -> 9 A53
rule_114: A53 -> 1
rule_115: A53 -> 13 3 A54 2
rule_116: A53 -> 15 A56 6
rule_117: A54 -> 1
rule_118: A54 -> 12 5 A55
rule_119: A54 -> 16 9 A53
rule_120: A55 -> 1
rule_121: A56 -> 1
rule_122: A56 -> 14 A53 8
rule_123: A57 -> 1
rule_124: A57 -> 3 A59
rule_125: A57 -> 5 A60 4
rule_126: A57 -> 7 A61 6
rule_127: A57 -> 9 A58 8
rule_128: A58 -> 1
rule_129: A58 -> 13 3 A59 2
rule_130: A58 -> 15 A61 6
rule_131: A59 -> 1
rule_132: A59 -> 12 5 A60
rule_133: A59 -> 16 9 A58 8
rule_134: A60 -> 1
rule_135: A61 -> 1
rule_136: A61 -> 14 A58
rule_137: A62 -> 1
rule_138: A62 -> 3 A64
rule_139: A62 -> 5 A65
rule_140: A62 -> 7 A66 6
rule_141: A62 -> 9 A63 8
rule_142: A63 -> 1
rule_143: A63 -> 13 A64 2
rule_144: A63 -> 15 A66 6
rule_145: A64 -> 1
rule_146: A64 -> 12 5 A65
rule_147: A64 -> 16 A63
rule_148: A65 -> 1
rule_149: A66 -> 1
rule_150: A66 -> 14 9 A63
rule_151: A67 -> A37
rule_152: A67 -> A42
rule_153: A67 -> A47
rule_154: A68 -> A37
rule_155: A69 -> 1
rule_156: A69 -> 3 A71
rule_157: A69 -> 5 A72
rule_158: A69 -> 7 A73
rule_159: A69 -> 9 A70
rule_160: A70 -> 1
rule_161: A70 -> 13 3 A71
rule_162: A70 -> 15 7 A73 6
rule_163: A71 -> 1
rule_164: A71 -> 12 A72 4
rule_165: A71 -> 16
rule_166: A71 -> 16 9 A70
rule_167: A72 -> 1
rule_168: A73 -> 1
rule_169: A73 -> 14
rule_170: A73 -> 14 A70 8
rule_171: A74 -> A91 A75 A92
rule_172: A74 -> A103 A107
rule_173: A75 -> A76
rule_174: A75 -> A81
rule_175: A76 -> 1
rule_176: A76 -> 3 A78 2
rule_177: A76 -> 5 A80 4
rule_178: A76 -> 7 A79
rule_179: A76 -> 9 A77 8
rule_180: A77 -> 1
rule_181: A78 -> 1
rule_182: A79 -> 1
rule_183: A80 -> 1
rule_184: A81 -> 1
rule_185: A81 -> 3 A83 2
rule_186: A81 -> 5 A85 4
rule_187: A81 -> 7 A84 6
rule_188: A81 -> 9 A82 8
rule_189: A82 -> 1
rule_190: A83 -> 1
rule_191: A84 -> 1
rule_192: A85 -> 1
rule_193: A86 -> 1
rule_194: A86 -> 3 A88
rule_195: A86 -> 5 A90
rule_196: A86 -> 7 A89 6
rule_197: A86 -> 9 A87 8
rule_198: A87 -> 1
rule_199: A88 -> 1
rule_200: A89 -> 1
rule_201: A90 -> 1
rule_202: A91 -> A76
rule_203: A91 -> A86
rule_204: A92 -> A76
rule_205: A92 -> A98
rule_206: A93 -> 1
rule_207: A93 -> 3 A95
rule_208: A93 -> 5 A97 4
rule_209: A93 -> 7 A96 6
rule_210: A93 -> 9 A94
rule_211: A94 -> 1
rule_212: A95 -> 1
rule_213: A96 -> 1
rule_214: A97 -> 1
rule_215: A98 -> 1
rule_216: A98 -> 3 A99 2
rule_217: A98 -> 5 A102 4
rule_218: A98 -> 7 A101 6
rule_219: A98 -> 9 A100 8
rule_220: A99 -> 1
rule_221: A100 -> 1
rule_222: A101 -> 1
rule_223: A102 -> 1
rule_224: A103 -> A67 A104
rule_225: A103 -> A106 A105
rule_226: A104 -> A37 19
rule_227: A104 -> A42
rule_228: A105 -> A37
rule_229: A105 -> A42
rule_230: A105 -> A47
rule_231: A106 -> A37
rule_232: A106 -> A42
rule_233: A106 -> A57
rule_234: A107 -> A37
rule_235: A107 -> A62
rule_236: A108 -> A29 A109 A110
rule_237: A108 -> A67 A111 A112
rule_238: A109 -> A14
rule_239: A109 -> A19
rule_240: A110 -> A4
rule_241: A110 -> A24
rule_242: A110 -> A31
rule_243: A111 -> A37 21
rule_244: A111 -> A52
rule_245: A111 -> A57
rule_246: A112 -> A37 22
rule_247: A112 -> A42
rule_248: A112 -> A69
rule_249: A113 -> A116 A114 23 A115
rule_250: A114 -> 1
rule_251: A114 -> 24
rule_252: A115 -> A37 25
rule_253: A116 -> 1
rule_254: A116 -> 3 A117 2
rule_255: A116 -> 5 A118
rule_256: A116 -> 7 A120 6
rule_257: A116 -> 9 A119
rule_258: A117 -> 1
rule_259: A117 -> 12
rule_260: A117 -> 12 A118 4
rule_261: A117 -> 16 9 A119
rule_262: A118 -> 1
rule_263: A119 -> 1
rule_264: A119 -> 13 3 A117
rule_265: A119 -> 15 A120
rule_266: A120 -> 1
rule_267: A120 -> 14 9 A119 8
rule_268: A121 -> A42 A114 26 A122
rule_269: A122 -> A37 27
rule_270: A122 -> A42
rule_271: A123 -> A37 A114 28 A124
rule_272: A123 -> A76 A125 A126
rule_273: A124 -> A37
rule_274: A124 -> A57
rule_275: A125 -> 1
rule_276: A126 -> A93

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