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 31 terminals, ranging from 1 to 31.
Each terminal symbol occurs in the target sequence at least once.

There are 39 non-terminal symbols, ranging from A1 to A39.

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

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