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 34 terminals, ranging from 1 to 34.
Each terminal symbol occurs in the target sequence at least once.

There are 77 non-terminal symbols, ranging from A1 to A77.

Production rules:
rule_1: A1 -> A68 A58 A51 A41 A31 A18 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A15
rule_4: A3 -> 1
rule_5: A3 -> A10 A8
rule_6: A4 -> 1
rule_7: A5 -> 1
rule_8: A6 -> 1
rule_9: A6 -> 3
rule_10: A7 -> 1
rule_11: A8 -> 4 A9
rule_12: A9 -> A5
rule_13: A9 -> A6 5
rule_14: A9 -> A7
rule_15: A10 -> 1
rule_16: A10 -> A13 A11
rule_17: A11 -> 6 A12
rule_18: A12 -> A6 7
rule_19: A13 -> 1
rule_20: A14 -> 1
rule_21: A15 -> A16 2
rule_22: A15 -> A17
rule_23: A16 -> A7
rule_24: A16 -> 9
rule_25: A17 -> A5
rule_26: A17 -> A7
rule_27: A18 -> 1
rule_28: A18 -> 1 A19 A25
rule_29: A18 -> A23 A19 A24
rule_30: A18 -> A30 A19 A29
rule_31: A19 -> 1
rule_32: A20 -> 1
rule_33: A20 -> 11
rule_34: A21 -> 1
rule_35: A22 -> 1
rule_36: A23 -> 1
rule_37: A23 -> 12
rule_38: A23 -> 13
rule_39: A24 -> 14 10
rule_40: A25 -> A26
rule_41: A26 -> A21
rule_42: A26 -> A23
rule_43: A27 -> A6 1 A28
rule_44: A28 -> 15 8
rule_45: A29 -> 14 10
rule_46: A30 -> A27 13
rule_47: A31 -> 1
rule_48: A31 -> A14 A13 A35
rule_49: A31 -> A19 A32 A37
rule_50: A31 -> A20 A32 A34
rule_51: A32 -> 1
rule_52: A33 -> A20
rule_53: A34 -> 18
rule_54: A35 -> A36
rule_55: A36 -> A4
rule_56: A36 -> A7
rule_57: A37 -> A38
rule_58: A37 -> A39
rule_59: A37 -> A40
rule_60: A38 -> 19
rule_61: A38 -> A20 18
rule_62: A39 -> A21
rule_63: A39 -> A22
rule_64: A39 -> A23
rule_65: A40 -> A20
rule_66: A41 -> 1
rule_67: A41 -> A23 A42 A44
rule_68: A41 -> A32 A42 A45
rule_69: A41 -> A48 A42 A47
rule_70: A42 -> 1
rule_71: A42 -> A19 A43
rule_72: A43 -> 19 A33
rule_73: A44 -> 21
rule_74: A45 -> A46 20
rule_75: A46 -> A22
rule_76: A46 -> A23
rule_77: A47 -> 21 20
rule_78: A48 -> A49 13
rule_79: A49 -> A6 A13 A50
rule_80: A50 -> 6 7
rule_81: A51 -> 1
rule_82: A51 -> 1 A52 A57 22
rule_83: A51 -> A54 23 25 A52 27 22
rule_84: A51 -> A54 A52 22
rule_85: A51 -> A55 A52
rule_86: A51 -> A56 A52 22
rule_87: A52 -> 1
rule_88: A52 -> 1 A53
rule_89: A53 -> A54 24
rule_90: A53 -> A56
rule_91: A54 -> 1
rule_92: A54 -> 25
rule_93: A54 -> 26
rule_94: A55 -> 1
rule_95: A56 -> 1
rule_96: A57 -> 23
rule_97: A57 -> A54
rule_98: A58 -> 1
rule_99: A58 -> 1 A59 A64
rule_100: A58 -> A61 A59 28
rule_101: A58 -> A62 A59 28
rule_102: A58 -> A65 A59 32 28
rule_103: A59 -> 1
rule_104: A59 -> 1 A60
rule_105: A59 -> 1 A63
rule_106: A60 -> A61
rule_107: A60 -> A62
rule_108: A61 -> 1
rule_109: A62 -> 1
rule_110: A62 -> 29
rule_111: A63 -> A62
rule_112: A64 -> A62 32
rule_113: A65 -> A66 29
rule_114: A66 -> A6 A10 A67
rule_115: A67 -> 4
rule_116: A68 -> 1
rule_117: A68 -> 1 A70 A71
rule_118: A68 -> A23 A70 A69
rule_119: A68 -> A76 A70 A75
rule_120: A69 -> 34 33
rule_121: A70 -> 1
rule_122: A71 -> A72
rule_123: A71 -> A73
rule_124: A71 -> A74
rule_125: A72 -> A22
rule_126: A72 -> A23
rule_127: A73 -> A23
rule_128: A74 -> A22
rule_129: A74 -> A23 34
rule_130: A75 -> 34
rule_131: A76 -> A77 12
rule_132: A77 -> A62 1 30

Target sequence:
1 1 30 31 12 1 34 33 3 1 6 1 7 4 5 29 1 32 28 26 1 23 24 25 1 27 22 12 1 19 11 17 21 20 1 1 18 16 3 1 15 8 13 1 14 10 1 1 9 2
"""