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 36 terminals, ranging from 1 to 36.
Each terminal symbol occurs in the target sequence at least once.

There are 57 non-terminal symbols, ranging from A1 to A57.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> A57
rule_3: A3 -> 4 A4
rule_4: A3 -> 5 A5
rule_5: A3 -> 7 A6
rule_6: A4 -> 1
rule_7: A4 -> 2
rule_8: A4 -> 3
rule_9: A5 -> 1
rule_10: A5 -> 2
rule_11: A5 -> 3
rule_12: A6 -> 1
rule_13: A6 -> 2
rule_14: A6 -> 3
rule_15: A6 -> 6
rule_16: A7 -> A8 A9
rule_17: A8 -> 10
rule_18: A9 -> 4 A10
rule_19: A9 -> 5 A11
rule_20: A9 -> 7 A12
rule_21: A10 -> 1
rule_22: A10 -> 2
rule_23: A10 -> 3
rule_24: A11 -> 1
rule_25: A11 -> 2
rule_26: A11 -> 3
rule_27: A12 -> 1
rule_28: A12 -> 2
rule_29: A12 -> 3
rule_30: A13 -> 11
rule_31: A13 -> A14 A15
rule_32: A14 -> 12
rule_33: A15 -> 4 A16
rule_34: A15 -> 5 A17
rule_35: A15 -> 7 A18
rule_36: A16 -> 1
rule_37: A16 -> 2
rule_38: A16 -> 3
rule_39: A17 -> 1
rule_40: A17 -> 2
rule_41: A17 -> 3
rule_42: A18 -> 1
rule_43: A18 -> 2
rule_44: A18 -> 3
rule_45: A19 -> A20 A3
rule_46: A20 -> 13
rule_47: A21 -> A22 A23
rule_48: A22 -> 14
rule_49: A23 -> 15 A24
rule_50: A23 -> 17 A25
rule_51: A24 -> 1
rule_52: A24 -> 2
rule_53: A24 -> 3
rule_54: A25 -> 1
rule_55: A25 -> 2
rule_56: A25 -> 3
rule_57: A25 -> 16
rule_58: A26 -> A27 A28
rule_59: A27 -> 18
rule_60: A28 -> 15 A29
rule_61: A28 -> 17 A30
rule_62: A29 -> 1
rule_63: A29 -> 2
rule_64: A29 -> 3
rule_65: A30 -> 1
rule_66: A30 -> 2
rule_67: A30 -> 3
rule_68: A31 -> A32
rule_69: A31 -> A33
rule_70: A31 -> A34
rule_71: A31 -> A35
rule_72: A31 -> A36
rule_73: A31 -> A37
rule_74: A31 -> A38
rule_75: A31 -> A39
rule_76: A31 -> A40
rule_77: A31 -> A41
rule_78: A31 -> A42
rule_79: A31 -> A43
rule_80: A32 -> 8
rule_81: A33 -> 8
rule_82: A34 -> 8
rule_83: A35 -> 8
rule_84: A36 -> 8
rule_85: A36 -> 20
rule_86: A37 -> 8
rule_87: A38 -> 8
rule_88: A39 -> 8
rule_89: A40 -> 8
rule_90: A41 -> 8
rule_91: A42 -> 8
rule_92: A43 -> 8
rule_93: A44 -> 21 A45
rule_94: A44 -> 22 A46
rule_95: A45 -> 9
rule_96: A46 -> 9
rule_97: A46 -> 23
rule_98: A47 -> A48 A44 A13 A31
rule_99: A48 -> 24 A7
rule_100: A49 -> A50 A44 A19 A31
rule_101: A50 -> 25 A13
rule_102: A51 -> 26 A49
rule_103: A52 -> 28 A47 A53
rule_104: A53 -> 19
rule_105: A54 -> 32
rule_106: A54 -> A55 A52 A21 A51
rule_107: A55 -> 33 A26
rule_108: A56 -> A54 30 A54 35
rule_109: A57 -> 36 A56 34

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