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 23 terminals, ranging from 1 to 23.
Each terminal symbol occurs in the target sequence at least once.

There are 30 non-terminal symbols, ranging from A1 to A30.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> A29 20 A30 A28
rule_3: A3 -> 2 A4
rule_4: A3 -> 3 A5
rule_5: A4 -> 1
rule_6: A5 -> 1
rule_7: A6 -> A7 A8
rule_8: A7 -> 5
rule_9: A8 -> 7 A9
rule_10: A8 -> 8 A10
rule_11: A9 -> 1
rule_12: A9 -> 6
rule_13: A10 -> 1
rule_14: A11 -> A12 A13
rule_15: A12 -> 9
rule_16: A13 -> 7 A14
rule_17: A13 -> 8 A15
rule_18: A14 -> 1
rule_19: A15 -> 1
rule_20: A15 -> 10
rule_21: A16 -> A17 A3
rule_22: A17 -> 11
rule_23: A18 -> A19 A20
rule_24: A19 -> 12
rule_25: A20 -> 2 A22
rule_26: A20 -> 3 A21
rule_27: A21 -> 1
rule_28: A21 -> 13
rule_29: A22 -> 1
rule_30: A23 -> 14
rule_31: A24 -> 15 A25
rule_32: A25 -> 4
rule_33: A26 -> 16
rule_34: A26 -> A27 A24 A18 A23
rule_35: A27 -> 17 A16
rule_36: A28 -> A26 18
rule_37: A29 -> A26 19
rule_38: A30 -> 21 A11 A6

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