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 36 terminals, ranging from 1 to 36.
Each terminal symbol occurs in the target sequence at least once.

There are 47 non-terminal symbols, ranging from A1 to A47.

Production rules:
rule_1: A1 -> A45 A42 A40 A38 A33 A28 A25 A19 A13 A2
rule_2: A2 -> A5 A4
rule_3: A2 -> A8 A7
rule_4: A2 -> A10 A9
rule_5: A2 -> A12 A11
rule_6: A3 -> 4
rule_7: A4 -> 5 2 1
rule_8: A5 -> 6
rule_9: A6 -> 11
rule_10: A7 -> 12 1
rule_11: A8 -> 13
rule_12: A9 -> 14 2 1
rule_13: A10 -> 15
rule_14: A11 -> 18 17 1
rule_15: A12 -> 19
rule_16: A13 -> A15 A14
rule_17: A14 -> 16 22 20
rule_18: A15 -> 23
rule_19: A16 -> 24
rule_20: A17 -> 25
rule_21: A18 -> 26
rule_22: A19 -> A3 A20
rule_23: A19 -> A5 A21
rule_24: A19 -> A6 A22
rule_25: A19 -> A8 A23
rule_26: A19 -> A10 A24
rule_27: A20 -> 3 22 21
rule_28: A21 -> 5 22
rule_29: A22 -> 10 22 21
rule_30: A23 -> 12 22
rule_31: A24 -> 14 21
rule_32: A25 -> A16 A26
rule_33: A25 -> A18 A27
rule_34: A26 -> 7 2 1
rule_35: A27 -> 9 28 1
rule_36: A28 -> A3 A29
rule_37: A28 -> A5 A30
rule_38: A28 -> A6 A31
rule_39: A28 -> A10 A32
rule_40: A29 -> 3 22
rule_41: A30 -> 5 21
rule_42: A31 -> 10 22 21
rule_43: A32 -> 14
rule_44: A33 -> A3 A34
rule_45: A33 -> A5 A35
rule_46: A33 -> A6 A36
rule_47: A33 -> A8 A37
rule_48: A34 -> 3 22 30 21
rule_49: A35 -> 5 22 21
rule_50: A36 -> 10 22 21
rule_51: A37 -> 12 22 21
rule_52: A38 -> A5 A39
rule_53: A39 -> 5 32 33 31
rule_54: A40 -> A16 A41
rule_55: A41 -> 7 1
rule_56: A42 -> A15 A43
rule_57: A42 -> A17 A44
rule_58: A43 -> 16 2
rule_59: A44 -> 8 35
rule_60: A45 -> A3 A46
rule_61: A45 -> A8 A47
rule_62: A46 -> 3 2
rule_63: A47 -> 12 36

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