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 32 terminals, ranging from 1 to 32.
Each terminal symbol occurs in the target sequence at least once.

There are 50 non-terminal symbols, ranging from A1 to A50.

Production rules:
rule_1: A1 -> A20 A48 A50 A24 A48 28 A49 A4 A45 A47 A7 A45 23 A46 A38 A30 A2
rule_2: A2 -> A13 A3 A14
rule_3: A2 -> A28 A15 A29
rule_4: A3 -> A7
rule_5: A3 -> A10
rule_6: A4 -> 1
rule_7: A4 -> 4 A5 3
rule_8: A4 -> 7 A6
rule_9: A4 -> 9 1
rule_10: A5 -> 1
rule_11: A5 -> 5 7 A6 6
rule_12: A6 -> 1
rule_13: A6 -> 2
rule_14: A7 -> 1
rule_15: A7 -> 4 A8 3
rule_16: A7 -> 7 A9
rule_17: A8 -> 1
rule_18: A8 -> 5
rule_19: A8 -> 5 A9 6
rule_20: A9 -> 1
rule_21: A10 -> 1
rule_22: A10 -> 4 A11 3
rule_23: A10 -> 7 A12
rule_24: A11 -> 1
rule_25: A11 -> 5 A12
rule_26: A12 -> 1
rule_27: A13 -> A7 10
rule_28: A14 -> A7
rule_29: A15 -> A16
rule_30: A15 -> A20
rule_31: A15 -> A24
rule_32: A16 -> 1
rule_33: A16 -> 4 A18
rule_34: A16 -> 7 A17 6
rule_35: A16 -> 9 A19 8
rule_36: A17 -> 1
rule_37: A17 -> 12
rule_38: A17 -> 15 A19 8
rule_39: A18 -> 1
rule_40: A18 -> 13 7 A17 6
rule_41: A19 -> 1
rule_42: A19 -> 14 7 A17
rule_43: A20 -> 1
rule_44: A20 -> 4 A23
rule_45: A20 -> 7 A22
rule_46: A20 -> 9 A21 8
rule_47: A21 -> 1
rule_48: A21 -> 14
rule_49: A21 -> 14 7 A22 6
rule_50: A22 -> 1
rule_51: A22 -> 15 9 A21 8
rule_52: A23 -> 1
rule_53: A23 -> 13
rule_54: A23 -> 13 A22 6
rule_55: A24 -> 1
rule_56: A24 -> 4 A26 3
rule_57: A24 -> 7 A25 6
rule_58: A24 -> 9 A27 8
rule_59: A25 -> 1
rule_60: A25 -> 15
rule_61: A25 -> 15 A27
rule_62: A26 -> 1
rule_63: A26 -> 13 A25 6
rule_64: A27 -> 1
rule_65: A27 -> 14 A25
rule_66: A28 -> A20 16
rule_67: A28 -> A24
rule_68: A29 -> A16
rule_69: A29 -> A20
rule_70: A30 -> A13 A31 A32
rule_71: A30 -> A33 A37
rule_72: A31 -> A4
rule_73: A31 -> A7
rule_74: A32 -> A7
rule_75: A33 -> A28 A34
rule_76: A33 -> A36 A35
rule_77: A34 -> A16
rule_78: A34 -> A20
rule_79: A34 -> A24
rule_80: A35 -> A16
rule_81: A35 -> A20 18
rule_82: A36 -> A20
rule_83: A36 -> A24
rule_84: A37 -> A20 20
rule_85: A38 -> A13 A39 A40
rule_86: A38 -> A41 A44
rule_87: A39 -> A4 21
rule_88: A40 -> A4 22
rule_89: A41 -> A28 A42
rule_90: A41 -> A36 A43
rule_91: A42 -> A20
rule_92: A43 -> A16
rule_93: A43 -> A20
rule_94: A44 -> A16
rule_95: A45 -> 1
rule_96: A45 -> 24
rule_97: A46 -> A7
rule_98: A46 -> 25
rule_99: A47 -> A4
rule_100: A47 -> A7
rule_101: A47 -> 27
rule_102: A48 -> 1
rule_103: A48 -> 29
rule_104: A49 -> A24
rule_105: A49 -> 30
rule_106: A50 -> A16
rule_107: A50 -> A24
rule_108: A50 -> 32

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