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

Production rules:
rule_1: A1 -> A44 A38 A36 A33 A29 A25 A22 A19 A9 A2
rule_2: A2 -> A4 A3
rule_3: A2 -> A6 A5
rule_4: A3 -> 4 2 1
rule_5: A4 -> 5
rule_6: A5 -> 10 2 9
rule_7: A6 -> 11
rule_8: A7 -> 15
rule_9: A8 -> 17
rule_10: A9 -> A11 A10
rule_11: A9 -> A13 A12
rule_12: A9 -> A15 A14
rule_13: A10 -> 6 18
rule_14: A11 -> 21
rule_15: A12 -> 13
rule_16: A13 -> 22
rule_17: A14 -> 12
rule_18: A15 -> 23
rule_19: A16 -> 24
rule_20: A17 -> 25
rule_21: A18 -> 26
rule_22: A19 -> A16 A20
rule_23: A19 -> A18 A21
rule_24: A20 -> 7 28
rule_25: A21 -> 8 28 27
rule_26: A22 -> A15 A23
rule_27: A22 -> A16 A24
rule_28: A23 -> 12 2
rule_29: A24 -> 7 1
rule_30: A25 -> A16 A26
rule_31: A25 -> A17 A27
rule_32: A25 -> A18 A28
rule_33: A26 -> 7 2 1
rule_34: A27 -> 3 2
rule_35: A28 -> 8 1
rule_36: A29 -> A11 A30
rule_37: A29 -> A17 A31
rule_38: A29 -> A18 A32
rule_39: A30 -> 6 2
rule_40: A31 -> 3 2 32 1
rule_41: A32 -> 8 2 1
rule_42: A33 -> A4 A34
rule_43: A33 -> A8 A35
rule_44: A34 -> 4 28 33 27
rule_45: A35 -> 16 28 27
rule_46: A36 -> A8 A37
rule_47: A37 -> 16 19
rule_48: A38 -> A11 A39
rule_49: A38 -> A13 A40
rule_50: A38 -> A16 A41
rule_51: A38 -> A17 A42
rule_52: A38 -> A18 A43
rule_53: A39 -> 6 18
rule_54: A40 -> 13 19 35 18
rule_55: A41 -> 7 18
rule_56: A42 -> 3
rule_57: A43 -> 8 19 18
rule_58: A44 -> A7 A45
rule_59: A45 -> 14 28

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