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 27 terminals, ranging from 1 to 27.
Each terminal symbol occurs in the target sequence at least once.

There are 141 non-terminal symbols, ranging from A1 to A141.

Production rules:
rule_1: A1 -> A138 A135 A95 A90 A74 A2
rule_2: A2 -> A29 A3 A30
rule_3: A2 -> A67 A36 A68
rule_4: A3 -> A14
rule_5: A4 -> 1
rule_6: A4 -> 3 A5 2
rule_7: A4 -> 5 A7 4
rule_8: A4 -> 7 A6 6
rule_9: A4 -> 9 A8 8
rule_10: A5 -> 1
rule_11: A6 -> 1
rule_12: A7 -> 1
rule_13: A8 -> 1
rule_14: A9 -> 1
rule_15: A9 -> 3 A12 2
rule_16: A9 -> 5 A10 4
rule_17: A9 -> 7 A11 6
rule_18: A9 -> 9 A13
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
rule_25: A14 -> 5 A17
rule_26: A14 -> 7 A16 6
rule_27: A14 -> 9 A18 8
rule_28: A15 -> 1
rule_29: A16 -> 1
rule_30: A17 -> 1
rule_31: A18 -> 1
rule_32: A19 -> 1
rule_33: A19 -> 3 A23
rule_34: A19 -> 5 A21 4
rule_35: A19 -> 7 A22 6
rule_36: A19 -> 9 A20
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 A27 4
rule_44: A24 -> 7 A26 6
rule_45: A24 -> 9 A28 8
rule_46: A25 -> 1
rule_47: A26 -> 1
rule_48: A27 -> 1
rule_49: A28 -> 1
rule_50: A29 -> A4
rule_51: A29 -> A19
rule_52: A30 -> A4
rule_53: A30 -> A31
rule_54: A31 -> 1
rule_55: A31 -> 3 A34 2
rule_56: A31 -> 5 A32 4
rule_57: A31 -> 7 A33 6
rule_58: A31 -> 9 A35 8
rule_59: A32 -> 1
rule_60: A33 -> 1
rule_61: A34 -> 1
rule_62: A35 -> 1
rule_63: A36 -> A37
rule_64: A36 -> A42 12
rule_65: A36 -> A52
rule_66: A36 -> A57
rule_67: A37 -> 1
rule_68: A37 -> 3 A38 2
rule_69: A37 -> 5 A40
rule_70: A37 -> 7 A41 6
rule_71: A37 -> 9 A39 8
rule_72: A38 -> 1
rule_73: A39 -> 1
rule_74: A39 -> 11 5 A40 4
rule_75: A40 -> 1
rule_76: A41 -> 1
rule_77: A42 -> 1
rule_78: A42 -> 3 A44
rule_79: A42 -> 5 A45
rule_80: A42 -> 7 A46
rule_81: A42 -> 9 A43 8
rule_82: A43 -> 1
rule_83: A43 -> 11
rule_84: A43 -> 11 5 A45
rule_85: A44 -> 1
rule_86: A45 -> 1
rule_87: A46 -> 1
rule_88: A47 -> 1
rule_89: A47 -> 3 A49
rule_90: A47 -> 5 A50 4
rule_91: A47 -> 7 A51 6
rule_92: A47 -> 9 A48
rule_93: A48 -> 1
rule_94: A48 -> 11 A50
rule_95: A49 -> 1
rule_96: A50 -> 1
rule_97: A51 -> 1
rule_98: A52 -> 1
rule_99: A52 -> 3 A54 2
rule_100: A52 -> 5 A55 4
rule_101: A52 -> 7 A56 6
rule_102: A52 -> 9 A53 8
rule_103: A53 -> 1
rule_104: A53 -> 11 A55
rule_105: A54 -> 1
rule_106: A55 -> 1
rule_107: A56 -> 1
rule_108: A57 -> 1
rule_109: A57 -> 3 A59
rule_110: A57 -> 5 A60 4
rule_111: A57 -> 7 A61
rule_112: A57 -> 9 A58 8
rule_113: A58 -> 1
rule_114: A58 -> 11 A60
rule_115: A59 -> 1
rule_116: A60 -> 1
rule_117: A61 -> 1
rule_118: A62 -> 1
rule_119: A62 -> 3 A64 2
rule_120: A62 -> 5 A65 4
rule_121: A62 -> 7 A66 6
rule_122: A62 -> 9 A63 8
rule_123: A63 -> 1
rule_124: A63 -> 11 5 A65
rule_125: A64 -> 1
rule_126: A65 -> 1
rule_127: A66 -> 1
rule_128: A67 -> A42 13
rule_129: A67 -> A47
rule_130: A67 -> A52
rule_131: A67 -> A62
rule_132: A68 -> A37
rule_133: A68 -> A42
rule_134: A68 -> A57
rule_135: A68 -> A69 14
rule_136: A69 -> 1
rule_137: A69 -> 3 A71
rule_138: A69 -> 5 A73 4
rule_139: A69 -> 7 A72 6
rule_140: A69 -> 9 A70 8
rule_141: A70 -> 1
rule_142: A70 -> 11 5 A73 4
rule_143: A71 -> 1
rule_144: A72 -> 1
rule_145: A73 -> 1
rule_146: A73 -> 10
rule_147: A74 -> A87 A89
rule_148: A75 -> 1
rule_149: A75 -> 3 A78 2
rule_150: A75 -> 5 A77 4
rule_151: A75 -> 9 A76
rule_152: A76 -> 1
rule_153: A76 -> 15
rule_154: A77 -> 1
rule_155: A78 -> 1
rule_156: A79 -> 1
rule_157: A79 -> 3 A82
rule_158: A79 -> 5 A81 4
rule_159: A79 -> 7 6
rule_160: A79 -> 9 A80 8
rule_161: A80 -> 1
rule_162: A81 -> 1
rule_163: A82 -> 1
rule_164: A83 -> 1
rule_165: A83 -> 3 A86 2
rule_166: A83 -> 5 A85 4
rule_167: A83 -> 7
rule_168: A83 -> 9 A84 8
rule_169: A84 -> 1
rule_170: A85 -> 1
rule_171: A86 -> 1
rule_172: A87 -> A67 A88
rule_173: A88 -> A42 16
rule_174: A88 -> A47
rule_175: A88 -> A52
rule_176: A89 -> A37
rule_177: A89 -> A42 17
rule_178: A89 -> A69
rule_179: A90 -> A29 A91 A92
rule_180: A90 -> A67 A93 A94
rule_181: A91 -> A4
rule_182: A91 -> A9
rule_183: A91 -> A19
rule_184: A92 -> A9
rule_185: A92 -> A24
rule_186: A93 -> A37
rule_187: A93 -> A42
rule_188: A93 -> A47
rule_189: A94 -> A42 19
rule_190: A94 -> A62
rule_191: A95 -> A123 A96 A97
rule_192: A95 -> A130 A128 A129
rule_193: A96 -> 1
rule_194: A97 -> A108
rule_195: A97 -> A113
rule_196: A97 -> A118
rule_197: A98 -> 1
rule_198: A98 -> 3 A101 2
rule_199: A98 -> 5 A99
rule_200: A98 -> 7 A100 6
rule_201: A98 -> 9 A102
rule_202: A99 -> 1
rule_203: A100 -> 1
rule_204: A100 -> 22 3 A101
rule_205: A101 -> 1
rule_206: A102 -> 1
rule_207: A103 -> 1
rule_208: A103 -> 3 A106 2
rule_209: A103 -> 5 A104 4
rule_210: A103 -> 7 A105
rule_211: A103 -> 9 A107
rule_212: A104 -> 1
rule_213: A105 -> 1
rule_214: A105 -> 22 3 A106 2
rule_215: A106 -> 1
rule_216: A107 -> 1
rule_217: A108 -> 1
rule_218: A108 -> 3 A109 2
rule_219: A108 -> 5 A111 4
rule_220: A108 -> 7 A110 6
rule_221: A108 -> 9 A112 8
rule_222: A109 -> 1
rule_223: A109 -> 21
rule_224: A110 -> 1
rule_225: A110 -> 22 3 A109 2
rule_226: A111 -> 1
rule_227: A112 -> 1
rule_228: A113 -> 1
rule_229: A113 -> 3 A115 2
rule_230: A113 -> 5 A116
rule_231: A113 -> 7 A114 6
rule_232: A113 -> 9 A117
rule_233: A114 -> 1
rule_234: A114 -> 22 3 A115 2
rule_235: A115 -> 1
rule_236: A116 -> 1
rule_237: A117 -> 1
rule_238: A118 -> 1
rule_239: A118 -> 3 A121 2
rule_240: A118 -> 5 A119
rule_241: A118 -> 7 A120 6
rule_242: A118 -> 9 A122
rule_243: A119 -> 1
rule_244: A120 -> 1
rule_245: A120 -> 22 A121 2
rule_246: A121 -> 1
rule_247: A121 -> 21 A120
rule_248: A122 -> 1
rule_249: A123 -> 1
rule_250: A123 -> 3 A125
rule_251: A123 -> 5 A126 4
rule_252: A123 -> 7 A124
rule_253: A123 -> 9 A127
rule_254: A124 -> 1
rule_255: A124 -> 22
rule_256: A124 -> 22 A125 2
rule_257: A125 -> 1
rule_258: A126 -> 1
rule_259: A127 -> 1
rule_260: A128 -> 1
rule_261: A129 -> A69
rule_262: A130 -> 1
rule_263: A130 -> 3 A131 2
rule_264: A130 -> 5 A133 4
rule_265: A130 -> 7 A134
rule_266: A130 -> 9 A132 8
rule_267: A131 -> 1
rule_268: A132 -> 1
rule_269: A132 -> 11 A133 4
rule_270: A133 -> 1
rule_271: A133 -> 10 A132
rule_272: A134 -> 1
rule_273: A135 -> A42 A128 A136
rule_274: A135 -> A103 A96 A137
rule_275: A136 -> A42
rule_276: A136 -> A69
rule_277: A137 -> A108
rule_278: A138 -> A75 A139 A140
rule_279: A138 -> A98 A96 A141
rule_280: A139 -> 1
rule_281: A140 -> A75 27
rule_282: A140 -> A79
rule_283: A140 -> A83
rule_284: A141 -> A103
rule_285: A141 -> A108
rule_286: A141 -> A118

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