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 31 terminals, ranging from 1 to 31.
Each terminal symbol occurs in the target sequence at least once.

There are 54 non-terminal symbols, ranging from A1 to A54.

Production rules:
rule_1: A1 -> A20 A52 A54 A24 A52 A53 A4 A49 25 A51 A7 A49 A50 A42 A34 A2
rule_2: A2 -> A13 A3 A14
rule_3: A2 -> A32 A15 A33
rule_4: A3 -> A4
rule_5: A3 -> A7
rule_6: A3 -> A10
rule_7: A4 -> 1
rule_8: A4 -> 4 A5
rule_9: A4 -> 7 A6 6
rule_10: A4 -> 9
rule_11: A5 -> 1
rule_12: A5 -> 5 A6 6
rule_13: A6 -> 1
rule_14: A6 -> 2
rule_15: A6 -> 2 A5
rule_16: A7 -> 1
rule_17: A7 -> 4 A8 3
rule_18: A7 -> 7 A9 6
rule_19: A7 -> 9 1
rule_20: A8 -> 1
rule_21: A8 -> 5
rule_22: A8 -> 5 7 A9
rule_23: A9 -> 1
rule_24: A10 -> 1
rule_25: A10 -> 4 A11
rule_26: A10 -> 7 A12 6
rule_27: A10 -> 9
rule_28: A11 -> 1
rule_29: A11 -> 5 7 A12 6
rule_30: A12 -> 1
rule_31: A13 -> A7
rule_32: A14 -> A4
rule_33: A14 -> A7
rule_34: A15 -> A20 14
rule_35: A15 -> A24
rule_36: A16 -> 1
rule_37: A16 -> 4 A18
rule_38: A16 -> 7 A17
rule_39: A16 -> 9 A19
rule_40: A17 -> 1
rule_41: A17 -> 13 9 A19
rule_42: A18 -> 1
rule_43: A18 -> 11 7 A17 6
rule_44: A19 -> 1
rule_45: A20 -> 1
rule_46: A20 -> 4 A23 3
rule_47: A20 -> 7 A22
rule_48: A20 -> 9 A21
rule_49: A21 -> 1
rule_50: A21 -> 12
rule_51: A21 -> 12 A22
rule_52: A22 -> 1
rule_53: A22 -> 13 A21 8
rule_54: A23 -> 1
rule_55: A23 -> 11
rule_56: A23 -> 11 7 A22 6
rule_57: A24 -> 1
rule_58: A24 -> 4 A26 3
rule_59: A24 -> 7 A25 6
rule_60: A24 -> 9 A27
rule_61: A25 -> 1
rule_62: A25 -> 13
rule_63: A25 -> 13 9 A27
rule_64: A26 -> 1
rule_65: A26 -> 11 7 A25
rule_66: A27 -> 1
rule_67: A28 -> 1
rule_68: A28 -> 4 A30 3
rule_69: A28 -> 7 A29 6
rule_70: A28 -> 9 A31
rule_71: A29 -> 1
rule_72: A29 -> 13 A31 8
rule_73: A30 -> 1
rule_74: A30 -> 11 A29
rule_75: A31 -> 1
rule_76: A32 -> A24
rule_77: A33 -> A16
rule_78: A33 -> A20 16
rule_79: A33 -> A24
rule_80: A34 -> A13 A35 A36
rule_81: A34 -> A37 A41
rule_82: A35 -> A4
rule_83: A36 -> A7
rule_84: A37 -> A32 A38
rule_85: A37 -> A40 A39
rule_86: A38 -> A16
rule_87: A38 -> A20
rule_88: A38 -> A24
rule_89: A39 -> A16
rule_90: A39 -> A20
rule_91: A39 -> A24
rule_92: A40 -> A24 18
rule_93: A41 -> A24
rule_94: A42 -> A13 A43 A44
rule_95: A42 -> A45 A48
rule_96: A43 -> A4
rule_97: A43 -> A10
rule_98: A44 -> A4 21
rule_99: A45 -> A32 A46
rule_100: A45 -> A40 A47
rule_101: A46 -> A16
rule_102: A47 -> A16
rule_103: A47 -> A28
rule_104: A48 -> A20
rule_105: A48 -> A24
rule_106: A49 -> 1
rule_107: A49 -> 23
rule_108: A50 -> 24
rule_109: A51 -> A4
rule_110: A51 -> 26
rule_111: A52 -> 1
rule_112: A52 -> 28
rule_113: A53 -> A16
rule_114: A53 -> A24
rule_115: A54 -> A16
rule_116: A54 -> A20
rule_117: A54 -> 31

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