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 25 terminals, ranging from 1 to 25.
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 -> A28 A23 A17 A12 A2
rule_2: A2 -> 1
rule_3: A2 -> A8 A7 A5 A3
rule_4: A3 -> 3 2
rule_5: A3 -> A4 2
rule_6: A4 -> 4
rule_7: A5 -> 1
rule_8: A5 -> A6
rule_9: A6 -> 5
rule_10: A7 -> 1
rule_11: A7 -> A8 7
rule_12: A7 -> A10 13
rule_13: A8 -> 1
rule_14: A8 -> A9 8
rule_15: A9 -> 1
rule_16: A9 -> 1 10
rule_17: A10 -> 1
rule_18: A10 -> A11 14 15
rule_19: A11 -> 1
rule_20: A11 -> A8 7
rule_21: A12 -> 1
rule_22: A12 -> A7 A9 A15 A13
rule_23: A13 -> A14 16
rule_24: A13 -> 17 16
rule_25: A14 -> 5
rule_26: A15 -> 1
rule_27: A15 -> A16 9
rule_28: A16 -> 8
rule_29: A17 -> 1
rule_30: A17 -> A9 A22 A20 A18
rule_31: A18 -> 19 18
rule_32: A18 -> A19 18
rule_33: A19 -> 8
rule_34: A20 -> 1
rule_35: A20 -> A21
rule_36: A21 -> 20
rule_37: A22 -> 1
rule_38: A23 -> 1
rule_39: A23 -> A22 A27 A26 A24
rule_40: A24 -> 23
rule_41: A24 -> A25
rule_42: A25 -> 20
rule_43: A26 -> 1
rule_44: A27 -> 1
rule_45: A27 -> A7 5 6
rule_46: A28 -> 1
rule_47: A28 -> A27 A30 A29
rule_48: A29 -> 25 24
rule_49: A30 -> 1
rule_50: A30 -> A22

Target sequence:
1 10 11 8 9 4 7 14 15 12 13 5 6 1 20 21 1 25 24 1 1 1 23 22 1 1 1 19 18 1 1 1 17 16 1 1 1 3 2
"""