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 38 terminals, ranging from 1 to 38.
Each terminal symbol occurs in the target sequence at least once.

There are 49 non-terminal symbols, ranging from A1 to A49.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> A47 A49 38 A48 A46
rule_3: A3 -> A4 A5
rule_4: A4 -> 5
rule_5: A5 -> 3 A6
rule_6: A5 -> 4 A7
rule_7: A6 -> 1
rule_8: A6 -> 6
rule_9: A7 -> 1
rule_10: A8 -> A9 A10
rule_11: A9 -> 7
rule_12: A10 -> 3 A11
rule_13: A10 -> 4 A12
rule_14: A11 -> 1
rule_15: A12 -> 1
rule_16: A12 -> 8
rule_17: A13 -> A14 A15
rule_18: A14 -> 10
rule_19: A15 -> 9 A16
rule_20: A16 -> 1
rule_21: A16 -> 11
rule_22: A17 -> A18 A19
rule_23: A18 -> 12
rule_24: A19 -> 2 A20
rule_25: A20 -> 1
rule_26: A21 -> A22 A23
rule_27: A22 -> 15
rule_28: A23 -> 13 A24
rule_29: A23 -> 14 A25
rule_30: A24 -> 1
rule_31: A25 -> 1
rule_32: A25 -> 16
rule_33: A26 -> A27 A28
rule_34: A27 -> 17
rule_35: A28 -> 13 A29
rule_36: A28 -> 14 A30
rule_37: A29 -> 1
rule_38: A29 -> 18
rule_39: A30 -> 1
rule_40: A31 -> 19
rule_41: A32 -> 20
rule_42: A33 -> 21
rule_43: A34 -> A35 A33 A26 A32
rule_44: A35 -> 23 A21
rule_45: A36 -> 24
rule_46: A37 -> 25 26
rule_47: A38 -> 27
rule_48: A39 -> 28
rule_49: A40 -> 31 A31
rule_50: A41 -> A42 A40 A39 A3 A38
rule_51: A42 -> 32 A8
rule_52: A43 -> A44 A37 A13 A36
rule_53: A44 -> 33 A17
rule_54: A45 -> 34
rule_55: A46 -> A41
rule_56: A46 -> A45 35
rule_57: A47 -> A41
rule_58: A47 -> A43 36
rule_59: A47 -> A45
rule_60: A48 -> A41 37
rule_61: A49 -> A34

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