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 38 terminals, ranging from 1 to 38.
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 -> A2
rule_2: A2 -> 37 33 A58 A63 A8 A8 35 A62 34
rule_3: A2 -> 37 33 A58 A65 A8 36 A8 A64 33
rule_4: A2 -> 37 33 A58 A69 A8 A8 35 A68 33
rule_5: A2 -> 37 33 A58 A75 A8 36 A8 A74 34 33
rule_6: A2 -> 37 A58 A67 A8 36 A8 A66 33
rule_7: A2 -> 37 A58 A71 A8 36 A8 A70 33
rule_8: A2 -> 37 A58 A73 A8 36 38 A8 35 A72 34
rule_9: A2 -> 37 A58 A77 A8 36 A8 35 A76 34
rule_10: A3 -> A4 A5
rule_11: A4 -> 8
rule_12: A5 -> 7 A6
rule_13: A6 -> 1
rule_14: A6 -> 2
rule_15: A7 -> 9
rule_16: A8 -> A9 A10
rule_17: A8 -> 11
rule_18: A9 -> 12
rule_19: A10 -> 10 A11
rule_20: A11 -> 1
rule_21: A11 -> 2
rule_22: A11 -> 13
rule_23: A12 -> 14 A13
rule_24: A13 -> 1
rule_25: A13 -> 2
rule_26: A14 -> A15 A16
rule_27: A15 -> 15
rule_28: A16 -> 4 A17
rule_29: A16 -> 5 A18
rule_30: A17 -> 1
rule_31: A17 -> 2
rule_32: A17 -> 16
rule_33: A18 -> 1
rule_34: A18 -> 2
rule_35: A19 -> A20 A21
rule_36: A20 -> 17
rule_37: A21 -> 4 A22
rule_38: A21 -> 5 A23
rule_39: A22 -> 1
rule_40: A22 -> 2
rule_41: A23 -> 1
rule_42: A23 -> 2
rule_43: A23 -> 18
rule_44: A24 -> 19
rule_45: A25 -> 20
rule_46: A26 -> A41 A42 A40 A14 A27
rule_47: A27 -> 21 A25 22
rule_48: A27 -> A28
rule_49: A27 -> A29
rule_50: A27 -> A30
rule_51: A27 -> A31
rule_52: A27 -> A32
rule_53: A27 -> A33
rule_54: A27 -> A34
rule_55: A27 -> A35
rule_56: A27 -> A36
rule_57: A27 -> A37
rule_58: A27 -> A38
rule_59: A27 -> A39
rule_60: A28 -> 3
rule_61: A29 -> 3
rule_62: A30 -> 3
rule_63: A31 -> 3
rule_64: A32 -> 3
rule_65: A33 -> 3
rule_66: A34 -> 3
rule_67: A35 -> 3
rule_68: A36 -> 3
rule_69: A37 -> 3
rule_70: A38 -> 3
rule_71: A39 -> 3
rule_72: A40 -> 23 A24 24
rule_73: A41 -> 25 A19
rule_74: A42 -> 27 A24
rule_75: A43 -> A44
rule_76: A43 -> A45
rule_77: A43 -> A46
rule_78: A43 -> A47
rule_79: A43 -> A48
rule_80: A43 -> A49
rule_81: A43 -> A50
rule_82: A43 -> A51
rule_83: A43 -> A52
rule_84: A43 -> A53
rule_85: A43 -> A54
rule_86: A43 -> A55
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 -> 3
rule_94: A51 -> 3
rule_95: A52 -> 3
rule_96: A53 -> 3
rule_97: A54 -> 3
rule_98: A55 -> 3
rule_99: A56 -> 28 A57
rule_100: A57 -> 6
rule_101: A58 -> A59 A12
rule_102: A59 -> 29
rule_103: A60 -> A61 A56 A3 A43
rule_104: A61 -> 30 A7
rule_105: A62 -> A26 31
rule_106: A63 -> A58
rule_107: A63 -> A60
rule_108: A64 -> A26 31
rule_109: A65 -> A58
rule_110: A65 -> A60 32
rule_111: A66 -> A26 31
rule_112: A66 -> A60
rule_113: A67 -> A26
rule_114: A67 -> A60
rule_115: A68 -> A26 31
rule_116: A69 -> A60 32
rule_117: A70 -> A26
rule_118: A70 -> A58
rule_119: A70 -> A60
rule_120: A71 -> A26
rule_121: A71 -> A60 32
rule_122: A72 -> A26
rule_123: A73 -> A58
rule_124: A73 -> A60 32
rule_125: A74 -> A26 31
rule_126: A74 -> A58
rule_127: A74 -> A60
rule_128: A75 -> A26
rule_129: A75 -> A58
rule_130: A75 -> A60 32
rule_131: A76 -> A26
rule_132: A76 -> A58
rule_133: A77 -> A60 32

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