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 -> A19 A23 A27 A2 A15
rule_2: A2 -> A4 A5 A3
rule_3: A3 -> 1
rule_4: A4 -> A7 A10
rule_5: A5 -> 2
rule_6: A5 -> 3
rule_7: A5 -> A6 2
rule_8: A6 -> 4
rule_9: A6 -> 5
rule_10: A6 -> A5 4
rule_11: A6 -> A7 5
rule_12: A7 -> 6
rule_13: A7 -> A6 7
rule_14: A7 -> 7
rule_15: A7 -> 8
rule_16: A7 -> A8
rule_17: A8 -> A7 9
rule_18: A8 -> 9
rule_19: A9 -> A7 10
rule_20: A9 -> 10
rule_21: A10 -> 11
rule_22: A11 -> 12
rule_23: A11 -> A12 12
rule_24: A12 -> 13
rule_25: A12 -> A13
rule_26: A13 -> 14
rule_27: A13 -> A14
rule_28: A14 -> A13 15
rule_29: A14 -> 15
rule_30: A15 -> A17 A9 A16
rule_31: A16 -> 16
rule_32: A17 -> A6 A18
rule_33: A18 -> 17
rule_34: A19 -> A21 A11 A20
rule_35: A20 -> 18
rule_36: A21 -> A14 A22
rule_37: A22 -> 19
rule_38: A23 -> A25 A5 A24
rule_39: A24 -> 20
rule_40: A25 -> A8 A26
rule_41: A26 -> 21
rule_42: A27 -> A29 A7 A28
rule_43: A28 -> 22
rule_44: A29 -> A5 A30
rule_45: A30 -> 23

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