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 37 terminals, ranging from 1 to 37.
Each terminal symbol occurs in the target sequence at least once.

There are 41 non-terminal symbols, ranging from A1 to A41.

Production rules:
rule_1: A1 -> A26 A38 A30 A34 A18 A2 A22
rule_2: A2 -> A4 A5 A3
rule_3: A3 -> 1
rule_4: A4 -> A7 A11
rule_5: A5 -> 2
rule_6: A5 -> 3
rule_7: A5 -> A7 2
rule_8: A5 -> A10
rule_9: A6 -> 4
rule_10: A6 -> 5
rule_11: A6 -> A5
rule_12: A6 -> A7 4
rule_13: A7 -> A5 8
rule_14: A7 -> 6
rule_15: A7 -> A6
rule_16: A7 -> 7
rule_17: A7 -> 8
rule_18: A7 -> A8
rule_19: A7 -> 9
rule_20: A8 -> A7
rule_21: A8 -> A9 11
rule_22: A8 -> 10
rule_23: A8 -> 11
rule_24: A9 -> A8 12
rule_25: A9 -> 12
rule_26: A10 -> A5
rule_27: A10 -> 13
rule_28: A10 -> 14
rule_29: A11 -> 15
rule_30: A12 -> A13
rule_31: A12 -> A14 17
rule_32: A12 -> 16
rule_33: A12 -> 17
rule_34: A12 -> A17
rule_35: A12 -> 18
rule_36: A12 -> 19
rule_37: A13 -> A12
rule_38: A13 -> A14
rule_39: A13 -> A16
rule_40: A13 -> 20
rule_41: A13 -> 21
rule_42: A14 -> A12
rule_43: A14 -> A13 22
rule_44: A14 -> A15
rule_45: A14 -> 22
rule_46: A15 -> A14
rule_47: A16 -> A13 23
rule_48: A16 -> 23
rule_49: A17 -> A12
rule_50: A17 -> 24
rule_51: A17 -> 25
rule_52: A18 -> A20 A9 A19
rule_53: A19 -> 26
rule_54: A20 -> A6 A21
rule_55: A21 -> 27
rule_56: A22 -> A24 A10 A23
rule_57: A23 -> 28
rule_58: A24 -> A7 A25
rule_59: A25 -> 29
rule_60: A26 -> A28 A12 A27
rule_61: A27 -> 30
rule_62: A28 -> A16 A29
rule_63: A29 -> 31
rule_64: A30 -> A32 A8 A31
rule_65: A31 -> 32
rule_66: A32 -> A10 A33
rule_67: A33 -> 33
rule_68: A34 -> A36 A17 A35
rule_69: A35 -> 34
rule_70: A36 -> A16 A37
rule_71: A37 -> 35
rule_72: A38 -> A40 A12 A39
rule_73: A39 -> 36
rule_74: A40 -> A17 A41
rule_75: A41 -> 37

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