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 19 terminals, ranging from 1 to 19.
Each terminal symbol occurs in the target sequence at least once.

There are 25 non-terminal symbols, ranging from A1 to A25.

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

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