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 12 terminals, ranging from 1 to 12.
Each terminal symbol occurs in the target sequence at least once.

There are 22 non-terminal symbols, ranging from A1 to A22.

Production rules:
rule_1: A1 -> A12 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A7 2
rule_4: A2 -> A5 A3 2
rule_5: A2 -> A6 A3
rule_6: A2 -> A9 A3 2
rule_7: A3 -> 1
rule_8: A3 -> 1 A4
rule_9: A4 -> A6
rule_10: A5 -> 1
rule_11: A6 -> 1
rule_12: A6 -> 3
rule_13: A7 -> A5
rule_14: A7 -> A6
rule_15: A8 -> 1
rule_16: A9 -> A10 3
rule_17: A10 -> A11 1 5
rule_18: A10 -> A11 6
rule_19: A11 -> 1
rule_20: A11 -> 7
rule_21: A12 -> 1
rule_22: A12 -> 1 A13 A15 8
rule_23: A12 -> A11 A13 9 8
rule_24: A12 -> A16 A13 8
rule_25: A12 -> A20 A13
rule_26: A13 -> 1
rule_27: A13 -> 1 5 A14
rule_28: A14 -> A8
rule_29: A14 -> A11 6
rule_30: A15 -> A11
rule_31: A16 -> A17
rule_32: A16 -> A19
rule_33: A17 -> A18 1 10
rule_34: A18 -> 1
rule_35: A19 -> A8 1
rule_36: A20 -> A21 7
rule_37: A21 -> A22 1 11
rule_38: A21 -> A22 11
rule_39: A22 -> 1
rule_40: A22 -> 12

Target sequence:
12 1 10 11 7 1 5 1 6 9 8 3 1 4 2
"""