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 19 non-terminal symbols, ranging from A1 to A19.

Production rules:
rule_1: A1 -> A3 A18 18 A19 A7 A18 15 A17 A16 A2 A15
rule_2: A2 -> A3
rule_3: A2 -> A11
rule_4: A3 -> 2
rule_5: A3 -> 6 A4 5
rule_6: A3 -> 8 A5 7
rule_7: A3 -> 11 A6
rule_8: A4 -> 2
rule_9: A4 -> 3
rule_10: A4 -> 3 A5 7
rule_11: A4 -> 12 11 A6
rule_12: A5 -> 2
rule_13: A5 -> 4 A4 5
rule_14: A6 -> 2
rule_15: A6 -> 9 6 A4
rule_16: A7 -> 2
rule_17: A7 -> 6 A8
rule_18: A7 -> 8 A9 7
rule_19: A7 -> 11 A10
rule_20: A8 -> 2
rule_21: A8 -> 3 A9 7
rule_22: A8 -> 12
rule_23: A8 -> 12 A10
rule_24: A9 -> 2
rule_25: A9 -> 4 A8 5
rule_26: A10 -> 2
rule_27: A10 -> 9 6 A8
rule_28: A11 -> 2
rule_29: A11 -> 6 A13 5
rule_30: A11 -> 8 A14 7
rule_31: A11 -> 11 A12 10
rule_32: A12 -> 2
rule_33: A12 -> 9
rule_34: A12 -> 9 6 A13 5
rule_35: A13 -> 2
rule_36: A13 -> 3 8 A14
rule_37: A13 -> 12 11 A12 10
rule_38: A14 -> 2
rule_39: A14 -> 4
rule_40: A14 -> 4 6 A13 5
rule_41: A15 -> A3
rule_42: A15 -> A11 13
rule_43: A16 -> A3
rule_44: A16 -> A7 14
rule_45: A16 -> A11
rule_46: A17 -> A11
rule_47: A17 -> 16
rule_48: A18 -> 2
rule_49: A18 -> 17
rule_50: A19 -> A11 19

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