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 61 non-terminal symbols, ranging from A1 to A61.

Production rules:
rule_1: A1 -> A55 A48 A32 A17 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A3 A7
rule_4: A2 -> A4 A3 A6
rule_5: A2 -> A11 A3 A10
rule_6: A3 -> 1
rule_7: A4 -> 1
rule_8: A4 -> 3
rule_9: A5 -> 1
rule_10: A6 -> 4 2
rule_11: A7 -> A8 2
rule_12: A8 -> A4
rule_13: A8 -> A5
rule_14: A9 -> 1
rule_15: A9 -> 5
rule_16: A10 -> 4 2
rule_17: A11 -> A12
rule_18: A12 -> A13 1 A14
rule_19: A13 -> 1
rule_20: A14 -> 6
rule_21: A15 -> 1
rule_22: A16 -> 1
rule_23: A17 -> 1
rule_24: A17 -> 1 A18 A24 8
rule_25: A17 -> A19 A18
rule_26: A17 -> A20 A18 8
rule_27: A17 -> A22 A18 8
rule_28: A17 -> A27 A18
rule_29: A18 -> 1
rule_30: A18 -> 1 12 A23
rule_31: A19 -> 1
rule_32: A20 -> 1
rule_33: A21 -> 1
rule_34: A22 -> 1
rule_35: A22 -> 10
rule_36: A22 -> 11
rule_37: A23 -> A19
rule_38: A23 -> A21
rule_39: A23 -> A22 13
rule_40: A24 -> A20
rule_41: A24 -> A21
rule_42: A24 -> A22
rule_43: A25 -> 1
rule_44: A26 -> 1
rule_45: A27 -> A28
rule_46: A27 -> A30
rule_47: A28 -> A29 1 14 15
rule_48: A29 -> 1
rule_49: A29 -> 16
rule_50: A29 -> 17
rule_51: A30 -> A22
rule_52: A31 -> 1
rule_53: A32 -> 1
rule_54: A32 -> 1 A33 A41
rule_55: A32 -> A9 A33 A40
rule_56: A32 -> A46 A33 A45
rule_57: A33 -> 1
rule_58: A33 -> A36 A34
rule_59: A34 -> 19 A35
rule_60: A35 -> A9 20
rule_61: A35 -> A16
rule_62: A36 -> 1
rule_63: A36 -> A39 A37
rule_64: A37 -> 21 A38
rule_65: A38 -> A9
rule_66: A39 -> 1
rule_67: A40 -> 23 18
rule_68: A41 -> A42
rule_69: A42 -> A9 23
rule_70: A42 -> A13
rule_71: A43 -> 1
rule_72: A43 -> 1 14 A44
rule_73: A44 -> A26
rule_74: A44 -> A29
rule_75: A44 -> A31
rule_76: A45 -> 23 18
rule_77: A46 -> A47
rule_78: A47 -> A29 A43 25
rule_79: A47 -> A29 A43 25
rule_80: A48 -> 1
rule_81: A48 -> 1 A39 A50
rule_82: A48 -> 1 A49 A52
rule_83: A48 -> A25 A49
rule_84: A48 -> A26 A49
rule_85: A48 -> A29 A49 27
rule_86: A48 -> A53 A49 27
rule_87: A49 -> 1
rule_88: A50 -> A51
rule_89: A51 -> A15
rule_90: A51 -> A16
rule_91: A52 -> A26
rule_92: A52 -> A29
rule_93: A53 -> A54
rule_94: A54 -> A22
rule_95: A54 -> A22 1
rule_96: A55 -> 1
rule_97: A55 -> A26 A56 28
rule_98: A55 -> A29 A56 29 28
rule_99: A55 -> A49 A56 A58 28
rule_100: A55 -> A59 A56 28
rule_101: A56 -> 1
rule_102: A56 -> A43 A57
rule_103: A57 -> A29 25
rule_104: A58 -> A26
rule_105: A58 -> A29 29
rule_106: A59 -> A60 16
rule_107: A60 -> A9 A39 A61
rule_108: A61 -> 21 22

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