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 29 terminals, ranging from 1 to 29.
Each terminal symbol occurs in the target sequence at least once.

There are 53 non-terminal symbols, ranging from A1 to A53.

Production rules:
rule_1: A1 -> A48 A39 A34 A25 A19 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A12 A16 8
rule_4: A2 -> A4 A3
rule_5: A2 -> A13 A12
rule_6: A2 -> A14 A12
rule_7: A2 -> A18 A12 12
rule_8: A3 -> 1
rule_9: A4 -> 1
rule_10: A4 -> 2
rule_11: A5 -> 1
rule_12: A6 -> 1
rule_13: A6 -> A10 A7
rule_14: A7 -> A8
rule_15: A7 -> A9 4
rule_16: A8 -> 1
rule_17: A9 -> 1
rule_18: A9 -> 5
rule_19: A10 -> 1
rule_20: A10 -> 1 A11
rule_21: A11 -> A9 7
rule_22: A12 -> 1
rule_23: A13 -> 1
rule_24: A14 -> 1
rule_25: A14 -> 9
rule_26: A14 -> 10
rule_27: A15 -> A13
rule_28: A15 -> A14 11
rule_29: A16 -> 13
rule_30: A16 -> A13
rule_31: A16 -> A14 12
rule_32: A17 -> A14 11
rule_33: A17 -> A14 13 11
rule_34: A18 -> A17
rule_35: A19 -> 1
rule_36: A19 -> 1 A20 A22 14
rule_37: A19 -> A4 A20 15 14
rule_38: A19 -> A24 A20 14
rule_39: A20 -> 1
rule_40: A21 -> 1
rule_41: A21 -> 1 13 A15
rule_42: A22 -> A4
rule_43: A23 -> A8 A10 3
rule_44: A24 -> A23
rule_45: A25 -> 1
rule_46: A25 -> A12 A27 A29 16
rule_47: A25 -> A13 A27
rule_48: A25 -> A14 A27 17 16
rule_49: A25 -> A31 A27
rule_50: A25 -> A32 A27
rule_51: A26 -> 1
rule_52: A27 -> 1
rule_53: A27 -> A12 A28
rule_54: A28 -> A14
rule_55: A29 -> A14
rule_56: A30 -> A9 A6 19 18
rule_57: A31 -> A30
rule_58: A32 -> A33
rule_59: A33 -> A8 A6
rule_60: A34 -> 1
rule_61: A34 -> 1 A35 A37 20
rule_62: A34 -> A9 A35 21
rule_63: A34 -> A38 A35 20
rule_64: A35 -> 1
rule_65: A35 -> 1 A11
rule_66: A35 -> A6 19 A36
rule_67: A36 -> A8
rule_68: A36 -> A9
rule_69: A37 -> 6
rule_70: A37 -> A9
rule_71: A38 -> A9 1 6 7
rule_72: A38 -> A9 1 7
rule_73: A39 -> 1
rule_74: A39 -> A4 A40 23
rule_75: A39 -> A5 A40 22
rule_76: A39 -> A20 A40 A43
rule_77: A39 -> A44 A40 22
rule_78: A39 -> A46 A40 22
rule_79: A40 -> 1
rule_80: A40 -> A20 A41
rule_81: A40 -> A26 24 A42
rule_82: A41 -> A4
rule_83: A42 -> A4
rule_84: A42 -> A5
rule_85: A43 -> A4
rule_86: A44 -> A45
rule_87: A45 -> A14 A21 26
rule_88: A45 -> A14 A21 26
rule_89: A46 -> A47
rule_90: A47 -> A13 A21
rule_91: A48 -> 1
rule_92: A48 -> A14 A49 29
rule_93: A48 -> A27 A49 A51 28
rule_94: A48 -> A52 A49 29 28
rule_95: A49 -> 1
rule_96: A49 -> A21 27 A50
rule_97: A50 -> A14
rule_98: A51 -> A14
rule_99: A52 -> A53 10
rule_100: A53 -> A4 A26
rule_101: A53 -> A4 A26 25

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