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 24 terminals, ranging from 1 to 24.
Each terminal symbol occurs in the target sequence at least once.

There are 32 non-terminal symbols, ranging from A1 to A32.

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

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