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 26 terminals, ranging from 1 to 26.
Each terminal symbol occurs in the target sequence at least once.

There are 47 non-terminal symbols, ranging from A1 to A47.

Production rules:
rule_1: A1 -> A19 A45 A47 A22 A45 22 A46 A4 A42 20 A44 A7 A42 A43 A36 A30 A2
rule_2: A2 -> A13 A3 A14
rule_3: A2 -> A28 A15 A29
rule_4: A3 -> A4
rule_5: A3 -> A7 7
rule_6: A3 -> A10
rule_7: A4 -> 1
rule_8: A4 -> 3 A5 2
rule_9: A4 -> 6 A6 5
rule_10: A5 -> 1
rule_11: A5 -> 4 A6 5
rule_12: A6 -> 1
rule_13: A7 -> 1
rule_14: A7 -> 3 A8
rule_15: A7 -> 6 A9
rule_16: A8 -> 1
rule_17: A8 -> 4
rule_18: A8 -> 4 6 A9 5
rule_19: A9 -> 1
rule_20: A10 -> 1
rule_21: A10 -> 3 A11 2
rule_22: A10 -> 6 A12 5
rule_23: A11 -> 1
rule_24: A11 -> 4 6 A12 5
rule_25: A12 -> 1
rule_26: A13 -> A4
rule_27: A13 -> A7 8
rule_28: A14 -> A7
rule_29: A15 -> A16
rule_30: A15 -> A25
rule_31: A16 -> 1
rule_32: A16 -> 3 A18
rule_33: A16 -> 6 A17 5
rule_34: A17 -> 1
rule_35: A18 -> 1
rule_36: A18 -> 10 6 A17 5
rule_37: A19 -> 1
rule_38: A19 -> 3 A21
rule_39: A19 -> 6 A20 5
rule_40: A20 -> 1
rule_41: A21 -> 1
rule_42: A21 -> 10
rule_43: A21 -> 10 6 A20
rule_44: A22 -> 1
rule_45: A22 -> 3 A24 2
rule_46: A22 -> 6 A23 5
rule_47: A23 -> 1
rule_48: A23 -> 11
rule_49: A24 -> 1
rule_50: A24 -> 10 6 A23 5
rule_51: A25 -> 1
rule_52: A25 -> 3 A27 2
rule_53: A25 -> 6 A26 5
rule_54: A26 -> 1
rule_55: A27 -> 1
rule_56: A27 -> 10 6 A26
rule_57: A28 -> A16
rule_58: A28 -> A22
rule_59: A29 -> A16
rule_60: A29 -> A22
rule_61: A30 -> A31 A35
rule_62: A31 -> A28 A32
rule_63: A31 -> A34 A33
rule_64: A32 -> A16
rule_65: A32 -> A22
rule_66: A33 -> A22 12
rule_67: A34 -> A16
rule_68: A34 -> A19
rule_69: A34 -> A22 13
rule_70: A35 -> A22 14
rule_71: A36 -> A13 A37 A38
rule_72: A36 -> A39 A41
rule_73: A37 -> A4
rule_74: A37 -> A7
rule_75: A38 -> A4
rule_76: A39 -> A34 A40
rule_77: A40 -> A19
rule_78: A40 -> A22 15
rule_79: A40 -> A25
rule_80: A41 -> A22
rule_81: A42 -> 1
rule_82: A42 -> 18
rule_83: A43 -> A4
rule_84: A43 -> A7 19
rule_85: A44 -> A7
rule_86: A45 -> 1
rule_87: A45 -> 23
rule_88: A46 -> A16
rule_89: A46 -> A22
rule_90: A46 -> 24
rule_91: A47 -> A22

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