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 40 terminals, ranging from 1 to 40.
Each terminal symbol occurs in the target sequence at least once.

There are 45 non-terminal symbols, ranging from A1 to A45.

Production rules:
rule_1: A1 -> A40 A37 A34 A32 A27 A24 A20 A16 A10 A2
rule_2: A2 -> A7 A6
rule_3: A3 -> 8
rule_4: A4 -> 10
rule_5: A5 -> 13
rule_6: A6 -> 15 14 1
rule_7: A7 -> 16
rule_8: A8 -> 18
rule_9: A9 -> 21
rule_10: A10 -> A12 A11
rule_11: A11 -> 19
rule_12: A12 -> 27
rule_13: A13 -> 28
rule_14: A14 -> 29
rule_15: A15 -> 30
rule_16: A16 -> A7 A17
rule_17: A16 -> A8 A18
rule_18: A16 -> A9 A19
rule_19: A17 -> 15
rule_20: A18 -> 17 24
rule_21: A19 -> 20 31 25
rule_22: A20 -> A12 A21
rule_23: A20 -> A13 A22
rule_24: A20 -> A15 A23
rule_25: A21 -> 19 2 1
rule_26: A22 -> 6 4
rule_27: A23 -> 5 4 32 3
rule_28: A24 -> A3 A25
rule_29: A24 -> A5 A26
rule_30: A25 -> 7 26 33
rule_31: A26 -> 12
rule_32: A27 -> A3 A28
rule_33: A27 -> A4 A29
rule_34: A27 -> A8 A30
rule_35: A27 -> A9 A31
rule_36: A28 -> 7 26
rule_37: A29 -> 9 26 25
rule_38: A30 -> 17 24 34
rule_39: A31 -> 20
rule_40: A32 -> A4 A33
rule_41: A33 -> 9 36 37
rule_42: A34 -> A14 A35
rule_43: A34 -> A15 A36
rule_44: A35 -> 11 2 38 1
rule_45: A36 -> 5 3
rule_46: A37 -> A13 A38
rule_47: A37 -> A15 A39
rule_48: A38 -> 6 4 39
rule_49: A39 -> 5 4
rule_50: A40 -> A3 A41
rule_51: A40 -> A4 A42
rule_52: A40 -> A5 A43
rule_53: A40 -> A7 A44
rule_54: A40 -> A8 A45
rule_55: A41 -> 7 4 3
rule_56: A42 -> 9 4 3
rule_57: A43 -> 12 2 40
rule_58: A44 -> 15 2
rule_59: A45 -> 17 1

Target sequence:
13 12 2 40 1 28 6 4 39 3 29 11 2 38 1 10 9 36 37 35 18 17 24 34 23 8 7 26 33 25 30 5 4 32 3 21 20 26 31 25 27 19 24 22 23 16 15 2 14 1
"""