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 30 terminals, ranging from 1 to 30.
Each terminal symbol occurs in the target sequence at least once.

There are 51 non-terminal symbols, ranging from A1 to A51.

Production rules:
rule_1: A1 -> A45 A37 A33 A25 A18 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A11 A15 9
rule_4: A2 -> A12 A11 10
rule_5: A2 -> A16 A11 9
rule_6: A3 -> 1
rule_7: A4 -> 1
rule_8: A4 -> 2
rule_9: A4 -> 3
rule_10: A5 -> 1
rule_11: A5 -> A9 A6
rule_12: A6 -> A7 5
rule_13: A6 -> A8
rule_14: A7 -> 1
rule_15: A7 -> 6
rule_16: A8 -> 1
rule_17: A9 -> 1
rule_18: A9 -> 1 7 A10
rule_19: A10 -> A7 8
rule_20: A10 -> A8
rule_21: A11 -> 1
rule_22: A12 -> 1
rule_23: A12 -> 11
rule_24: A12 -> 12
rule_25: A13 -> 1
rule_26: A14 -> A12
rule_27: A14 -> A13
rule_28: A15 -> A12 10
rule_29: A15 -> 14
rule_30: A16 -> A17
rule_31: A17 -> A12 14
rule_32: A18 -> 1
rule_33: A18 -> 1 A19 A21
rule_34: A18 -> A4 A19
rule_35: A18 -> A22 A20
rule_36: A18 -> A24 A19 16 15
rule_37: A19 -> 1
rule_38: A20 -> 1
rule_39: A20 -> 1 A14
rule_40: A21 -> A4
rule_41: A22 -> A23
rule_42: A23 -> A7 A9
rule_43: A24 -> A23
rule_44: A25 -> 1
rule_45: A25 -> A4 A26
rule_46: A25 -> A11 A27 A29 17
rule_47: A25 -> A12 A27 18
rule_48: A25 -> A30 A27 18
rule_49: A26 -> 1
rule_50: A27 -> 1
rule_51: A27 -> A11 A28
rule_52: A28 -> A12
rule_53: A29 -> A12 18
rule_54: A30 -> A31 12
rule_55: A30 -> A32
rule_56: A31 -> A7 A5 20
rule_57: A31 -> A7 A5 20
rule_58: A32 -> A12 A11
rule_59: A33 -> 1
rule_60: A33 -> 1 A34 A36
rule_61: A33 -> A7 A34 22
rule_62: A34 -> 1
rule_63: A34 -> A5 19 A35
rule_64: A35 -> A7 20
rule_65: A35 -> A8
rule_66: A36 -> 7
rule_67: A36 -> A7
rule_68: A36 -> A8
rule_69: A37 -> 1
rule_70: A37 -> A4 A38 26 23
rule_71: A37 -> A19 A38 A40
rule_72: A37 -> A41 A38
rule_73: A37 -> A43 A38 26
rule_74: A38 -> 1
rule_75: A38 -> A26 A39
rule_76: A39 -> A3
rule_77: A39 -> A4 25
rule_78: A40 -> A4 26
rule_79: A41 -> A42
rule_80: A42 -> A13 A20
rule_81: A43 -> A44 2
rule_82: A44 -> A12 A20 28
rule_83: A45 -> 1
rule_84: A45 -> A12 A46 30 29
rule_85: A45 -> A27 A46 A48
rule_86: A45 -> A49 A46 30
rule_87: A46 -> 1
rule_88: A46 -> A20 27 A47
rule_89: A47 -> A12 28
rule_90: A47 -> A13
rule_91: A48 -> A12 30
rule_92: A49 -> A50 11
rule_93: A49 -> A51
rule_94: A50 -> A4 A26 25
rule_95: A51 -> A12 A27

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