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 48 terminals, ranging from 1 to 48.
Each terminal symbol occurs in the target sequence at least once.

There are 106 non-terminal symbols, ranging from A1 to A106.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 48 A106 A93 42 A94 A105
rule_3: A3 -> 9 A4
rule_4: A4 -> 1
rule_5: A4 -> 2
rule_6: A4 -> 3
rule_7: A4 -> 4
rule_8: A4 -> 5
rule_9: A4 -> 6
rule_10: A5 -> A6 A7
rule_11: A6 -> 15
rule_12: A7 -> 7 A8
rule_13: A7 -> 8 A9
rule_14: A8 -> 1
rule_15: A8 -> 2
rule_16: A8 -> 3
rule_17: A8 -> 4
rule_18: A8 -> 5
rule_19: A8 -> 6
rule_20: A9 -> 1
rule_21: A9 -> 2
rule_22: A9 -> 3
rule_23: A9 -> 4
rule_24: A9 -> 5
rule_25: A9 -> 6
rule_26: A10 -> A11 A12
rule_27: A11 -> 16
rule_28: A12 -> 7 A13
rule_29: A12 -> 8 A14
rule_30: A13 -> 1
rule_31: A13 -> 2
rule_32: A13 -> 3
rule_33: A13 -> 4
rule_34: A13 -> 5
rule_35: A13 -> 6
rule_36: A14 -> 1
rule_37: A14 -> 2
rule_38: A14 -> 3
rule_39: A14 -> 4
rule_40: A14 -> 5
rule_41: A14 -> 6
rule_42: A15 -> 19
rule_43: A16 -> A17 A18
rule_44: A17 -> 20
rule_45: A18 -> 17 A19
rule_46: A18 -> 18 A20
rule_47: A19 -> 1
rule_48: A19 -> 2
rule_49: A19 -> 3
rule_50: A19 -> 4
rule_51: A19 -> 5
rule_52: A19 -> 6
rule_53: A20 -> 1
rule_54: A20 -> 2
rule_55: A20 -> 3
rule_56: A20 -> 4
rule_57: A20 -> 5
rule_58: A20 -> 6
rule_59: A21 -> 21
rule_60: A21 -> A22 A23
rule_61: A22 -> 22
rule_62: A23 -> 17 A24
rule_63: A23 -> 18 A25
rule_64: A24 -> 1
rule_65: A24 -> 2
rule_66: A24 -> 3
rule_67: A24 -> 4
rule_68: A24 -> 5
rule_69: A24 -> 6
rule_70: A25 -> 1
rule_71: A25 -> 2
rule_72: A25 -> 3
rule_73: A25 -> 4
rule_74: A25 -> 5
rule_75: A25 -> 6
rule_76: A25 -> 23
rule_77: A26 -> A27 A28
rule_78: A27 -> 26
rule_79: A28 -> 24 A29
rule_80: A28 -> 25 A30
rule_81: A29 -> 1
rule_82: A29 -> 2
rule_83: A29 -> 3
rule_84: A29 -> 4
rule_85: A29 -> 5
rule_86: A29 -> 6
rule_87: A29 -> 27
rule_88: A30 -> 1
rule_89: A30 -> 2
rule_90: A30 -> 3
rule_91: A30 -> 4
rule_92: A30 -> 5
rule_93: A30 -> 6
rule_94: A31 -> A32 A33
rule_95: A32 -> 28
rule_96: A33 -> 24 A34
rule_97: A33 -> 25 A35
rule_98: A34 -> 1
rule_99: A34 -> 2
rule_100: A34 -> 3
rule_101: A34 -> 4
rule_102: A34 -> 5
rule_103: A34 -> 6
rule_104: A35 -> 1
rule_105: A35 -> 2
rule_106: A35 -> 3
rule_107: A35 -> 4
rule_108: A35 -> 5
rule_109: A35 -> 6
rule_110: A36 -> 29 A37
rule_111: A37 -> 1
rule_112: A37 -> 2
rule_113: A37 -> 3
rule_114: A37 -> 4
rule_115: A37 -> 5
rule_116: A37 -> 6
rule_117: A38 -> A39
rule_118: A38 -> A40
rule_119: A38 -> A41
rule_120: A38 -> A42
rule_121: A38 -> A43
rule_122: A38 -> A44
rule_123: A38 -> A45
rule_124: A38 -> A46
rule_125: A38 -> A47
rule_126: A38 -> A48
rule_127: A38 -> A49
rule_128: A38 -> A50
rule_129: A39 -> 10
rule_130: A39 -> 11
rule_131: A40 -> 10
rule_132: A40 -> 11
rule_133: A41 -> 10
rule_134: A41 -> 11
rule_135: A42 -> 10
rule_136: A42 -> 11
rule_137: A43 -> 10
rule_138: A43 -> 11
rule_139: A44 -> 10
rule_140: A44 -> 11
rule_141: A45 -> 10
rule_142: A45 -> 11
rule_143: A46 -> 10
rule_144: A46 -> 11
rule_145: A47 -> 10
rule_146: A47 -> 11
rule_147: A48 -> 10
rule_148: A48 -> 11
rule_149: A49 -> 10
rule_150: A49 -> 11
rule_151: A50 -> 10
rule_152: A50 -> 11
rule_153: A51 -> 31 A52
rule_154: A52 -> 13
rule_155: A52 -> 14
rule_156: A53 -> A54
rule_157: A53 -> A55
rule_158: A53 -> A56
rule_159: A53 -> A57
rule_160: A53 -> A58
rule_161: A53 -> A59
rule_162: A53 -> A60
rule_163: A53 -> A61
rule_164: A53 -> A62
rule_165: A53 -> A63
rule_166: A53 -> A64
rule_167: A53 -> A65
rule_168: A54 -> 10
rule_169: A54 -> 11
rule_170: A55 -> 10
rule_171: A55 -> 11
rule_172: A56 -> 10
rule_173: A56 -> 11
rule_174: A57 -> 10
rule_175: A57 -> 11
rule_176: A58 -> 10
rule_177: A58 -> 11
rule_178: A59 -> 10
rule_179: A59 -> 11
rule_180: A60 -> 10
rule_181: A60 -> 11
rule_182: A60 -> 32
rule_183: A61 -> 10
rule_184: A61 -> 11
rule_185: A62 -> 10
rule_186: A62 -> 11
rule_187: A63 -> 10
rule_188: A63 -> 11
rule_189: A64 -> 10
rule_190: A64 -> 11
rule_191: A65 -> 10
rule_192: A65 -> 11
rule_193: A66 -> 33 A67
rule_194: A67 -> 13
rule_195: A67 -> 14
rule_196: A67 -> 34
rule_197: A68 -> A69 A36
rule_198: A69 -> 35
rule_199: A70 -> A71 A85 A16 A72
rule_200: A71 -> 36 A21
rule_201: A72 -> A73
rule_202: A72 -> A74
rule_203: A72 -> A75
rule_204: A72 -> A76
rule_205: A72 -> A77
rule_206: A72 -> A78
rule_207: A72 -> A79
rule_208: A72 -> A80
rule_209: A72 -> A81
rule_210: A72 -> A82
rule_211: A72 -> A83
rule_212: A72 -> A84
rule_213: A73 -> 10
rule_214: A73 -> 11
rule_215: A74 -> 10
rule_216: A74 -> 11
rule_217: A75 -> 10
rule_218: A75 -> 11
rule_219: A76 -> 10
rule_220: A76 -> 11
rule_221: A77 -> 10
rule_222: A77 -> 11
rule_223: A78 -> 10
rule_224: A78 -> 11
rule_225: A79 -> 10
rule_226: A79 -> 11
rule_227: A80 -> 10
rule_228: A80 -> 11
rule_229: A81 -> 10
rule_230: A81 -> 11
rule_231: A82 -> 10
rule_232: A82 -> 11
rule_233: A83 -> 10
rule_234: A83 -> 11
rule_235: A84 -> 10
rule_236: A84 -> 11
rule_237: A85 -> 37 A86
rule_238: A86 -> 13
rule_239: A86 -> 14
rule_240: A87 -> A88 A51 A10 A38
rule_241: A88 -> 38 A5
rule_242: A89 -> A90 A66 A21 A53
rule_243: A90 -> 39 A15
rule_244: A91 -> A92 A3
rule_245: A92 -> 40
rule_246: A93 -> 41 A87
rule_247: A94 -> A95
rule_248: A95 -> A96 A99 A31 A97
rule_249: A96 -> 43 A26
rule_250: A97 -> 44 A70 A98
rule_251: A98 -> 12
rule_252: A99 -> 45 A89 A100
rule_253: A100 -> 30
rule_254: A101 -> 47
rule_255: A102 -> 47
rule_256: A103 -> 47
rule_257: A104 -> 47
rule_258: A105 -> A68
rule_259: A105 -> A87
rule_260: A105 -> A91
rule_261: A106 -> A68 A104
rule_262: A106 -> A87 A103
rule_263: A106 -> A89 A102
rule_264: A106 -> A91 A101

Target sequence:
48 40 9 1 47 41 38 15 7 2 31 13 16 8 3 10 42 43 26 24 27 45 39 19 33 34 22 18 23 32 30 28 25 4 44 36 21 37 14 20 17 5 11 12 35 29 6 46
"""