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 52 non-terminal symbols, ranging from A1 to A52.

Production rules:
rule_1: A1 -> A47 A41 A36 A28 A21 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A13 A18
rule_4: A2 -> A9 A6
rule_5: A2 -> A15 A13 9
rule_6: A2 -> A16 A13 13 9
rule_7: A2 -> A20 A13 9
rule_8: A3 -> 1
rule_9: A4 -> 1
rule_10: A4 -> 2
rule_11: A5 -> 1
rule_12: A5 -> 3
rule_13: A6 -> 1
rule_14: A6 -> 1 A12
rule_15: A6 -> A10 A7
rule_16: A7 -> A8 5
rule_17: A7 -> A9
rule_18: A8 -> 1
rule_19: A8 -> 6
rule_20: A9 -> 1
rule_21: A10 -> 1
rule_22: A10 -> 1 A11
rule_23: A11 -> A8 8
rule_24: A12 -> A8
rule_25: A13 -> 1
rule_26: A13 -> 1 A14
rule_27: A14 -> A16
rule_28: A15 -> 1
rule_29: A15 -> 10
rule_30: A16 -> 1
rule_31: A16 -> 11
rule_32: A17 -> A16 12
rule_33: A18 -> 14
rule_34: A18 -> A15
rule_35: A18 -> A16 13
rule_36: A19 -> A4 1
rule_37: A20 -> A19
rule_38: A21 -> 1
rule_39: A21 -> 1 A22 A24
rule_40: A21 -> A4 A22 15
rule_41: A21 -> A5 A22 16
rule_42: A21 -> A27 A22 16 15
rule_43: A22 -> 1
rule_44: A23 -> 1
rule_45: A23 -> 1 A17
rule_46: A24 -> A4
rule_47: A24 -> A5 16
rule_48: A25 -> A8 A10 5
rule_49: A26 -> A15 1
rule_50: A27 -> A25
rule_51: A27 -> A26
rule_52: A28 -> 1
rule_53: A28 -> A3 A29 A32
rule_54: A28 -> A13 A30 A33
rule_55: A28 -> A15 A30 17
rule_56: A28 -> A16 A30 17
rule_57: A28 -> A34 A30 17
rule_58: A29 -> 1
rule_59: A30 -> 1
rule_60: A30 -> A13 A31
rule_61: A31 -> A16
rule_62: A32 -> A4
rule_63: A33 -> A15
rule_64: A34 -> A35 10
rule_65: A35 -> A8 A6 20
rule_66: A36 -> 1
rule_67: A36 -> 1 A37 A39 21
rule_68: A36 -> A8 A37 22 21
rule_69: A36 -> A40 6 A37 22 21
rule_70: A37 -> 1
rule_71: A37 -> 1 A11
rule_72: A37 -> A6 A38
rule_73: A38 -> A8 20
rule_74: A38 -> A9
rule_75: A39 -> 7
rule_76: A39 -> A8
rule_77: A40 -> A8 7 8
rule_78: A41 -> 1
rule_79: A41 -> A4 A42
rule_80: A41 -> A22 A42 A44 23
rule_81: A41 -> A45 A42 24 23
rule_82: A42 -> 1
rule_83: A42 -> A29 25 A43
rule_84: A43 -> A4 26
rule_85: A44 -> A4 24
rule_86: A44 -> A5
rule_87: A45 -> A46 2
rule_88: A46 -> A16 A23 28
rule_89: A47 -> 1
rule_90: A47 -> A15 A48
rule_91: A47 -> A16 A48 30
rule_92: A47 -> A30 A48 A50 29
rule_93: A47 -> A51 A48 30 29
rule_94: A48 -> 1
rule_95: A48 -> A23 28 A49
rule_96: A49 -> A15
rule_97: A49 -> A16
rule_98: A50 -> A16 30
rule_99: A51 -> A52 11
rule_100: A52 -> A4 A29 26
rule_101: A52 -> A4 A29 26

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