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 86 non-terminal symbols, ranging from A1 to A86.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 48 A86 A75 A76 A77 A85
rule_3: A3 -> 6
rule_4: A4 -> 8 A5
rule_5: A4 -> 9 A6
rule_6: A5 -> 1
rule_7: A5 -> 2
rule_8: A5 -> 3
rule_9: A5 -> 4
rule_10: A5 -> 7
rule_11: A6 -> 1
rule_12: A6 -> 2
rule_13: A6 -> 3
rule_14: A6 -> 4
rule_15: A7 -> A8 A9
rule_16: A8 -> 14
rule_17: A9 -> 12 A10
rule_18: A9 -> 13 A11
rule_19: A10 -> 1
rule_20: A10 -> 2
rule_21: A10 -> 3
rule_22: A10 -> 4
rule_23: A11 -> 1
rule_24: A11 -> 2
rule_25: A11 -> 3
rule_26: A11 -> 4
rule_27: A12 -> A13 A14
rule_28: A13 -> 15
rule_29: A14 -> 12 A15
rule_30: A14 -> 13 A16
rule_31: A15 -> 1
rule_32: A15 -> 2
rule_33: A15 -> 3
rule_34: A15 -> 4
rule_35: A16 -> 1
rule_36: A16 -> 2
rule_37: A16 -> 3
rule_38: A16 -> 4
rule_39: A17 -> A18 A19
rule_40: A18 -> 16
rule_41: A19 -> 8 A20
rule_42: A19 -> 9 A21
rule_43: A20 -> 1
rule_44: A20 -> 2
rule_45: A20 -> 3
rule_46: A20 -> 4
rule_47: A21 -> 1
rule_48: A21 -> 2
rule_49: A21 -> 3
rule_50: A21 -> 4
rule_51: A22 -> A23 A4
rule_52: A23 -> 17
rule_53: A24 -> 18 A25
rule_54: A25 -> 1
rule_55: A25 -> 2
rule_56: A25 -> 3
rule_57: A25 -> 4
rule_58: A26 -> 21
rule_59: A26 -> A27 A28
rule_60: A27 -> 22
rule_61: A28 -> 19 A29
rule_62: A28 -> 20 A30
rule_63: A29 -> 1
rule_64: A29 -> 2
rule_65: A29 -> 3
rule_66: A29 -> 4
rule_67: A29 -> 23
rule_68: A30 -> 1
rule_69: A30 -> 2
rule_70: A30 -> 3
rule_71: A30 -> 4
rule_72: A31 -> 24
rule_73: A31 -> A32 A33
rule_74: A32 -> 25
rule_75: A33 -> 19 A34
rule_76: A33 -> 20 A35
rule_77: A34 -> 1
rule_78: A34 -> 2
rule_79: A34 -> 3
rule_80: A34 -> 4
rule_81: A35 -> 1
rule_82: A35 -> 2
rule_83: A35 -> 3
rule_84: A35 -> 4
rule_85: A35 -> 26
rule_86: A36 -> A37
rule_87: A36 -> A38
rule_88: A36 -> A39
rule_89: A36 -> A40
rule_90: A36 -> A41
rule_91: A36 -> A42
rule_92: A36 -> A43
rule_93: A36 -> A44
rule_94: A36 -> A45
rule_95: A36 -> A46
rule_96: A36 -> A47
rule_97: A36 -> A48
rule_98: A37 -> 5
rule_99: A38 -> 5
rule_100: A39 -> 5
rule_101: A39 -> 27
rule_102: A40 -> 5
rule_103: A41 -> 5
rule_104: A42 -> 5
rule_105: A43 -> 5
rule_106: A44 -> 5
rule_107: A45 -> 5
rule_108: A46 -> 5
rule_109: A47 -> 5
rule_110: A48 -> 5
rule_111: A49 -> 28 A50
rule_112: A50 -> 10
rule_113: A50 -> 11
rule_114: A51 -> 29 A3 30
rule_115: A51 -> A52
rule_116: A51 -> A53
rule_117: A51 -> A54
rule_118: A51 -> A55
rule_119: A51 -> A56
rule_120: A51 -> A57
rule_121: A51 -> A58
rule_122: A51 -> A59
rule_123: A51 -> A60
rule_124: A51 -> A61
rule_125: A51 -> A62
rule_126: A51 -> A63
rule_127: A52 -> 5
rule_128: A53 -> 5
rule_129: A54 -> 5
rule_130: A55 -> 5
rule_131: A56 -> 5
rule_132: A57 -> 5
rule_133: A58 -> 5
rule_134: A59 -> 5
rule_135: A60 -> 5
rule_136: A61 -> 5
rule_137: A62 -> 5
rule_138: A63 -> 5
rule_139: A64 -> 31 A65
rule_140: A64 -> 33 A66
rule_141: A65 -> 10
rule_142: A65 -> 11
rule_143: A65 -> 32
rule_144: A66 -> 10
rule_145: A66 -> 11
rule_146: A67 -> A68 A24
rule_147: A68 -> 34
rule_148: A69 -> A70 A64 A31 A51
rule_149: A70 -> 35 A26
rule_150: A71 -> A72 A49 A22 A36
rule_151: A71 -> A73 A64 A26 A51
rule_152: A72 -> 36 A17
rule_153: A73 -> 37 A31
rule_154: A74 -> 38
rule_155: A75 -> 39 A74
rule_156: A76 -> 40
rule_157: A77 -> A80
rule_158: A78 -> 41 A69 42
rule_159: A79 -> 43 A71
rule_160: A80 -> A81 A79 A12 A78
rule_161: A81 -> 45 A7
rule_162: A82 -> 47
rule_163: A83 -> 47
rule_164: A84 -> 47
rule_165: A85 -> A67
rule_166: A85 -> A71 46
rule_167: A86 -> A67 A83
rule_168: A86 -> A71 A82
rule_169: A86 -> A74 A84

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