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 38 terminals, ranging from 1 to 38.
Each terminal symbol occurs in the target sequence at least once.

There are 42 non-terminal symbols, ranging from A1 to A42.

Production rules:
rule_1: A1 -> A40 A36 A33 A29 A26 A22 A20 A17 A9 A2
rule_2: A2 -> A5 A4
rule_3: A2 -> A8 A7
rule_4: A3 -> 6
rule_5: A4 -> 10 2 9 1
rule_6: A5 -> 11
rule_7: A6 -> 15
rule_8: A7 -> 16 1
rule_9: A8 -> 17
rule_10: A9 -> A15 A14
rule_11: A10 -> 22
rule_12: A11 -> 23
rule_13: A12 -> 24
rule_14: A13 -> 25
rule_15: A14 -> 7 21 26
rule_16: A15 -> 27
rule_17: A16 -> 28
rule_18: A17 -> A10 A18
rule_19: A17 -> A12 A19
rule_20: A18 -> 19 30
rule_21: A19 -> 12 29
rule_22: A20 -> A13 A21
rule_23: A21 -> 18 32
rule_24: A22 -> A11 A23
rule_25: A22 -> A13 A24
rule_26: A22 -> A15 A25
rule_27: A23 -> 8
rule_28: A24 -> 18 2 1
rule_29: A25 -> 7 2 1
rule_30: A26 -> A10 A27
rule_31: A26 -> A15 A28
rule_32: A27 -> 19 2 34
rule_33: A28 -> 7
rule_34: A29 -> A3 A30
rule_35: A29 -> A5 A31
rule_36: A29 -> A8 A32
rule_37: A30 -> 3 29
rule_38: A31 -> 10
rule_39: A32 -> 16 30 35 29
rule_40: A33 -> A3 A34
rule_41: A33 -> A8 A35
rule_42: A34 -> 3 21 20
rule_43: A35 -> 16 21
rule_44: A36 -> A10 A37
rule_45: A36 -> A15 A38
rule_46: A36 -> A16 A39
rule_47: A37 -> 19 21 20
rule_48: A38 -> 7 21
rule_49: A39 -> 13 20
rule_50: A40 -> A3 A41
rule_51: A40 -> A6 A42
rule_52: A41 -> 3 29
rule_53: A42 -> 14 30 29

Target sequence:
15 14 30 38 29 28 13 21 37 20 6 3 21 36 20 17 16 30 35 29 22 19 2 34 1 23 8 5 33 4 25 18 2 32 1 24 12 30 31 29 27 7 21 26 20 11 10 2 9 1
"""