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 21 terminals, ranging from 1 to 21.
Each terminal symbol occurs in the target sequence at least once.

There are 24 non-terminal symbols, ranging from A1 to A24.

Production rules:
rule_1: A1 -> A17 A21 A2 A13
rule_2: A2 -> A8 A9 A7
rule_3: A3 -> 1
rule_4: A3 -> A4 1
rule_5: A4 -> 2
rule_6: A4 -> 3
rule_7: A4 -> A3
rule_8: A4 -> A5
rule_9: A5 -> 4
rule_10: A5 -> A4 4
rule_11: A5 -> 5
rule_12: A5 -> 6
rule_13: A5 -> A6 5
rule_14: A6 -> A5 7
rule_15: A6 -> 7
rule_16: A7 -> 8
rule_17: A8 -> A11 A12
rule_18: A9 -> 9
rule_19: A9 -> A9 9
rule_20: A9 -> 10
rule_21: A9 -> A10 10
rule_22: A10 -> A9
rule_23: A10 -> 11
rule_24: A10 -> A11
rule_25: A10 -> 12
rule_26: A10 -> 13
rule_27: A11 -> A10 14
rule_28: A11 -> 14
rule_29: A12 -> 15
rule_30: A13 -> A15 A6 A14
rule_31: A14 -> 16
rule_32: A15 -> A5 A16
rule_33: A16 -> 17
rule_34: A17 -> A19 A6 A18
rule_35: A18 -> 18
rule_36: A19 -> A3 A20
rule_37: A20 -> 19
rule_38: A21 -> A23 A10 A22
rule_39: A22 -> 20
rule_40: A23 -> A9 A24
rule_41: A24 -> 21

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