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 32 terminals, ranging from 1 to 32.
Each terminal symbol occurs in the target sequence at least once.

There are 73 non-terminal symbols, ranging from A1 to A73.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 31 27 A54 A61 A8 30 A8 A60 27
rule_3: A2 -> 31 27 A54 A63 A8 A8 A62
rule_4: A2 -> 31 27 A54 A65 A8 30 A8 29 A64
rule_5: A2 -> 31 27 A54 A71 A8 32 A8 29 A70 28 27
rule_6: A2 -> 31 A54 A59 A8 A8 29 A58 27
rule_7: A2 -> 31 A54 A67 A8 A8 29 A66 28 27
rule_8: A2 -> 31 A54 A69 A8 A8 A68 27
rule_9: A2 -> 31 A54 A73 A8 A8 A72 27
rule_10: A3 -> A4 A5
rule_11: A4 -> 6
rule_12: A5 -> 5 A6
rule_13: A6 -> 1
rule_14: A6 -> 2
rule_15: A7 -> 7
rule_16: A8 -> 9
rule_17: A8 -> A9 A10
rule_18: A9 -> 10
rule_19: A10 -> 8 A11
rule_20: A11 -> 1
rule_21: A11 -> 2
rule_22: A11 -> 11
rule_23: A12 -> A13 A14
rule_24: A13 -> 14
rule_25: A14 -> 12 A15
rule_26: A14 -> 13 A16
rule_27: A15 -> 1
rule_28: A15 -> 2
rule_29: A16 -> 1
rule_30: A16 -> 2
rule_31: A16 -> 15
rule_32: A17 -> A18 A19
rule_33: A18 -> 16
rule_34: A19 -> 12 A20
rule_35: A19 -> 13 A21
rule_36: A20 -> 1
rule_37: A20 -> 2
rule_38: A20 -> 17
rule_39: A21 -> 1
rule_40: A21 -> 2
rule_41: A22 -> A23
rule_42: A22 -> A24
rule_43: A22 -> A25
rule_44: A22 -> A26
rule_45: A22 -> A27
rule_46: A22 -> A28
rule_47: A22 -> A29
rule_48: A22 -> A30
rule_49: A22 -> A31
rule_50: A22 -> A32
rule_51: A22 -> A33
rule_52: A22 -> A34
rule_53: A23 -> 3
rule_54: A24 -> 3
rule_55: A25 -> 3
rule_56: A26 -> 3
rule_57: A27 -> 3
rule_58: A28 -> 3
rule_59: A29 -> 3
rule_60: A30 -> 3
rule_61: A31 -> 3
rule_62: A32 -> 3
rule_63: A32 -> 18
rule_64: A33 -> 3
rule_65: A34 -> 3
rule_66: A35 -> 19 A36
rule_67: A36 -> 4
rule_68: A36 -> 20
rule_69: A37 -> A38
rule_70: A37 -> A39
rule_71: A37 -> A40
rule_72: A37 -> A41
rule_73: A37 -> A42
rule_74: A37 -> A43
rule_75: A37 -> A44
rule_76: A37 -> A45
rule_77: A37 -> A46
rule_78: A37 -> A47
rule_79: A37 -> A48
rule_80: A37 -> A49
rule_81: A38 -> 3
rule_82: A39 -> 3
rule_83: A40 -> 3
rule_84: A41 -> 3
rule_85: A42 -> 3
rule_86: A43 -> 3
rule_87: A44 -> 3
rule_88: A45 -> 3
rule_89: A46 -> 3
rule_90: A47 -> 3
rule_91: A48 -> 3
rule_92: A49 -> 3
rule_93: A50 -> 21 A51
rule_94: A51 -> 4
rule_95: A52 -> A53 A35 A12 A22
rule_96: A53 -> 22 A17
rule_97: A54 -> A55 A14
rule_98: A55 -> 23
rule_99: A56 -> A57 A50 A3 A37
rule_100: A57 -> 24 A7
rule_101: A58 -> A52
rule_102: A58 -> A56
rule_103: A59 -> A52
rule_104: A59 -> A56
rule_105: A60 -> A56 25
rule_106: A61 -> A52
rule_107: A61 -> A54
rule_108: A61 -> A56
rule_109: A62 -> A52
rule_110: A62 -> A56
rule_111: A63 -> A52 26
rule_112: A64 -> A56 25
rule_113: A65 -> A52
rule_114: A65 -> A56
rule_115: A66 -> A56 25
rule_116: A67 -> A52
rule_117: A68 -> A52
rule_118: A68 -> A56 25
rule_119: A69 -> A52 26
rule_120: A69 -> A54
rule_121: A69 -> A56
rule_122: A70 -> A52
rule_123: A70 -> A56 25
rule_124: A71 -> A52 26
rule_125: A71 -> A56
rule_126: A72 -> A56 25
rule_127: A73 -> A52

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