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 27 terminals, ranging from 1 to 27.
Each terminal symbol occurs in the target sequence at least once.

There are 44 non-terminal symbols, ranging from A1 to A44.

Production rules:
rule_1: A1 -> A36 A27 A20 A15 A11 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A7 2
rule_4: A2 -> A5 A3
rule_5: A2 -> A6 A3 2
rule_6: A2 -> A8 A3 2
rule_7: A3 -> 1
rule_8: A3 -> 1 A4
rule_9: A4 -> A5
rule_10: A5 -> 1
rule_11: A5 -> 4
rule_12: A6 -> 1
rule_13: A7 -> A5
rule_14: A7 -> A6
rule_15: A8 -> A9
rule_16: A9 -> A10 1 5
rule_17: A9 -> A10 1 7 5
rule_18: A10 -> 1
rule_19: A10 -> 6
rule_20: A11 -> 1
rule_21: A11 -> 1 A12 A14 8
rule_22: A11 -> A13 A12 8
rule_23: A12 -> 1
rule_24: A13 -> 1
rule_25: A14 -> 9
rule_26: A15 -> 1
rule_27: A15 -> 1 A16 A19 10
rule_28: A15 -> A18 A16
rule_29: A16 -> 1
rule_30: A17 -> 1
rule_31: A17 -> 11
rule_32: A18 -> 1
rule_33: A18 -> 12
rule_34: A19 -> 13
rule_35: A20 -> 1
rule_36: A20 -> 1 A21 A24 14
rule_37: A20 -> A23 A21 14
rule_38: A20 -> A25 A21
rule_39: A21 -> 1
rule_40: A22 -> 1
rule_41: A23 -> 1
rule_42: A23 -> 16
rule_43: A24 -> A22
rule_44: A24 -> A23
rule_45: A25 -> A26 16
rule_46: A26 -> A18 1 17
rule_47: A26 -> A18 1 18
rule_48: A27 -> 1
rule_49: A27 -> 1 A28 A31
rule_50: A27 -> A29 A28 19
rule_51: A27 -> A30 A28 19
rule_52: A27 -> A34 A28 20
rule_53: A28 -> 1
rule_54: A29 -> 1
rule_55: A29 -> 21
rule_56: A30 -> 1
rule_57: A31 -> A29
rule_58: A32 -> 1
rule_59: A32 -> 1 7 A33
rule_60: A33 -> A10 5
rule_61: A34 -> A35
rule_62: A35 -> A10 A32
rule_63: A35 -> A10 A32 23
rule_64: A36 -> 1
rule_65: A36 -> 1 A37 A39 24
rule_66: A36 -> A17 A37 25
rule_67: A36 -> A18 A37
rule_68: A36 -> A40 A37 25 24
rule_69: A37 -> 1
rule_70: A37 -> 1 A38
rule_71: A38 -> A18
rule_72: A39 -> A17
rule_73: A40 -> A41
rule_74: A40 -> A42
rule_75: A41 -> A17 1
rule_76: A42 -> A10 A43 26
rule_77: A43 -> 1
rule_78: A43 -> A32 22 A44
rule_79: A44 -> A10

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