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 44 non-terminal symbols, ranging from A1 to A44.

Production rules:
rule_1: A1 -> A41 A38 A36 A33 A30 A26 A21 A18 A8 A2
rule_2: A2 -> A5 A4
rule_3: A3 -> 6
rule_4: A4 -> 10 2 1
rule_5: A5 -> 11
rule_6: A6 -> 14
rule_7: A7 -> 17
rule_8: A8 -> A10 A9
rule_9: A8 -> A14 A13
rule_10: A8 -> A17 A16
rule_11: A9 -> 7 18
rule_12: A10 -> 20
rule_13: A11 -> 21
rule_14: A12 -> 22
rule_15: A13 -> 12 19 18
rule_16: A14 -> 24
rule_17: A15 -> 25
rule_18: A16 -> 8
rule_19: A17 -> 26
rule_20: A18 -> A12 A19
rule_21: A18 -> A15 A20
rule_22: A19 -> 4 30 31
rule_23: A20 -> 3 30 29
rule_24: A21 -> A10 A22
rule_25: A21 -> A11 A23
rule_26: A21 -> A14 A24
rule_27: A21 -> A17 A25
rule_28: A22 -> 7 2 1
rule_29: A23 -> 15
rule_30: A24 -> 12 2
rule_31: A25 -> 8 2 32 1
rule_32: A26 -> A10 A27
rule_33: A26 -> A15 A28
rule_34: A26 -> A17 A29
rule_35: A27 -> 7
rule_36: A28 -> 3 2 1
rule_37: A29 -> 8 2 1
rule_38: A30 -> A11 A31
rule_39: A30 -> A15 A32
rule_40: A31 -> 15 1
rule_41: A32 -> 3 34
rule_42: A33 -> A3 A34
rule_43: A33 -> A6 A35
rule_44: A34 -> 5 35
rule_45: A35 -> 13 30 29
rule_46: A36 -> A6 A37
rule_47: A37 -> 13 19 36 18
rule_48: A38 -> A11 A39
rule_49: A38 -> A14 A40
rule_50: A39 -> 15 18
rule_51: A40 -> 12 18
rule_52: A41 -> A3 A42
rule_53: A41 -> A6 A43
rule_54: A41 -> A7 A44
rule_55: A42 -> 5 30 29
rule_56: A43 -> 13 30 29
rule_57: A44 -> 16 27

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