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 25 terminals, ranging from 1 to 25.
Each terminal symbol occurs in the target sequence at least once.

There are 41 non-terminal symbols, ranging from A1 to A41.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 24 A33 A31 A35 A3 23 A3 A34 21 A33
rule_3: A2 -> 24 A33 A31 A37 A3 23 A3 22 A36 A33
rule_4: A2 -> 24 A33 A31 A39 A3 23 A3 A38 A33
rule_5: A2 -> 24 A33 A31 A41 A3 A3 A40 21 A33
rule_6: A3 -> A4 A5
rule_7: A3 -> 6
rule_8: A4 -> 7
rule_9: A5 -> 5 A6
rule_10: A6 -> 1
rule_11: A6 -> 2
rule_12: A6 -> 8
rule_13: A7 -> 9 A8
rule_14: A8 -> 1
rule_15: A8 -> 2
rule_16: A9 -> A10 A11
rule_17: A10 -> 11
rule_18: A11 -> 10 A12
rule_19: A12 -> 1
rule_20: A12 -> 2
rule_21: A13 -> 12
rule_22: A14 -> A15
rule_23: A14 -> A16
rule_24: A14 -> A17
rule_25: A14 -> A18
rule_26: A14 -> A19
rule_27: A14 -> A20
rule_28: A14 -> A21
rule_29: A14 -> A22
rule_30: A14 -> A23
rule_31: A14 -> A24
rule_32: A14 -> A25
rule_33: A14 -> A26
rule_34: A15 -> 3
rule_35: A16 -> 3
rule_36: A17 -> 3
rule_37: A18 -> 3
rule_38: A19 -> 3
rule_39: A20 -> 3
rule_40: A21 -> 3
rule_41: A22 -> 3
rule_42: A23 -> 3
rule_43: A24 -> 3
rule_44: A25 -> 3
rule_45: A26 -> 3
rule_46: A27 -> 13 A28
rule_47: A28 -> 4
rule_48: A29 -> A30 A27 A9 A14
rule_49: A30 -> 14 A13
rule_50: A31 -> 15
rule_51: A31 -> A32 A7
rule_52: A32 -> 16
rule_53: A33 -> 17
rule_54: A33 -> 20
rule_55: A34 -> A31 18
rule_56: A35 -> A29 19
rule_57: A36 -> A29
rule_58: A36 -> A31
rule_59: A37 -> A29 19
rule_60: A38 -> A29
rule_61: A38 -> A31
rule_62: A39 -> A29 19
rule_63: A40 -> A31
rule_64: A41 -> A29 19

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