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 24 terminals, ranging from 1 to 24.
Each terminal symbol occurs in the target sequence at least once.

There are 36 non-terminal symbols, ranging from A1 to A36.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> 24 A32 A34
rule_3: A3 -> 2 A4
rule_4: A3 -> 3 A5
rule_5: A3 -> 4 A6
rule_6: A4 -> 1
rule_7: A5 -> 1
rule_8: A6 -> 1
rule_9: A6 -> 5
rule_10: A7 -> A8 A9
rule_11: A8 -> 6
rule_12: A9 -> 2 A11
rule_13: A9 -> 3 A10
rule_14: A9 -> 4 A12
rule_15: A10 -> 1
rule_16: A11 -> 1
rule_17: A12 -> 1
rule_18: A13 -> A14 A15
rule_19: A14 -> 7
rule_20: A15 -> 2 A17
rule_21: A15 -> 3 A16
rule_22: A15 -> 4 A18
rule_23: A16 -> 1
rule_24: A17 -> 1
rule_25: A17 -> 8
rule_26: A18 -> 1
rule_27: A19 -> A20 A21
rule_28: A20 -> 10
rule_29: A21 -> 9 A22
rule_30: A22 -> 1
rule_31: A22 -> 11
rule_32: A23 -> 12
rule_33: A24 -> 13
rule_34: A25 -> 14
rule_35: A26 -> A27 A25 A13 A24
rule_36: A27 -> 16 A7
rule_37: A28 -> A29 A3
rule_38: A29 -> 17
rule_39: A30 -> 18 A28 19
rule_40: A31 -> 20 A26 21
rule_41: A32 -> A33
rule_42: A33 -> 22
rule_43: A34 -> A35
rule_44: A35 -> A36 A31 A19 A30
rule_45: A36 -> 23 A23

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