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 36 terminals, ranging from 1 to 36.
Each terminal symbol occurs in the target sequence at least once.

There are 55 non-terminal symbols, ranging from A1 to A55.

Production rules:
rule_1: A1 -> A52 A50 A47 A45 A43 A39 A36 A33 A31 A26 A17 A10 A2
rule_2: A2 -> A6 A5 3
rule_3: A3 -> 1
rule_4: A4 -> 1
rule_5: A4 -> 2
rule_6: A5 -> 1
rule_7: A5 -> 4
rule_8: A6 -> 1
rule_9: A6 -> 7 A9
rule_10: A7 -> 1
rule_11: A7 -> A6 A8
rule_12: A8 -> 1
rule_13: A8 -> 6
rule_14: A9 -> 1
rule_15: A9 -> 1
rule_16: A9 -> 1
rule_17: A9 -> 1
rule_18: A9 -> 1
rule_19: A9 -> 1
rule_20: A9 -> 1
rule_21: A9 -> 1
rule_22: A9 -> 1
rule_23: A9 -> 1
rule_24: A9 -> 1
rule_25: A9 -> 1
rule_26: A9 -> 1
rule_27: A9 -> 1
rule_28: A9 -> 1
rule_29: A9 -> 1
rule_30: A9 -> 1
rule_31: A9 -> 1
rule_32: A9 -> 1
rule_33: A9 -> 1
rule_34: A9 -> 1
rule_35: A9 -> 1
rule_36: A9 -> 1
rule_37: A9 -> 1
rule_38: A9 -> A7
rule_39: A10 -> A13 A12 9
rule_40: A10 -> A16 A12
rule_41: A11 -> 1
rule_42: A11 -> 8
rule_43: A12 -> 1
rule_44: A12 -> 10
rule_45: A13 -> 1
rule_46: A13 -> 12 A15
rule_47: A14 -> 1
rule_48: A14 -> A13 A4 11
rule_49: A15 -> 1
rule_50: A15 -> 1
rule_51: A15 -> 1
rule_52: A15 -> 1
rule_53: A15 -> 1
rule_54: A15 -> 1
rule_55: A15 -> 1
rule_56: A15 -> 1
rule_57: A15 -> 1
rule_58: A15 -> 1
rule_59: A15 -> 1
rule_60: A15 -> 1
rule_61: A15 -> 1
rule_62: A15 -> 1
rule_63: A15 -> 1
rule_64: A15 -> 1
rule_65: A15 -> 1
rule_66: A15 -> 1
rule_67: A15 -> 1
rule_68: A15 -> 1
rule_69: A15 -> 1
rule_70: A15 -> 1
rule_71: A15 -> 1
rule_72: A15 -> 1
rule_73: A15 -> A14
rule_74: A16 -> 1
rule_75: A17 -> A6 A22
rule_76: A17 -> A19 A18 13
rule_77: A17 -> A24 A23
rule_78: A18 -> 1
rule_79: A18 -> 14
rule_80: A19 -> 1
rule_81: A19 -> 16 A21
rule_82: A20 -> 1
rule_83: A20 -> A19 A11 15
rule_84: A21 -> 1
rule_85: A21 -> 1
rule_86: A21 -> 1
rule_87: A21 -> 1
rule_88: A21 -> 1
rule_89: A21 -> 1
rule_90: A21 -> 1
rule_91: A21 -> 1
rule_92: A21 -> 1
rule_93: A21 -> 1
rule_94: A21 -> 1
rule_95: A21 -> 1
rule_96: A21 -> 1
rule_97: A21 -> 1
rule_98: A21 -> 1
rule_99: A21 -> 1
rule_100: A21 -> 1
rule_101: A21 -> 1
rule_102: A21 -> 1
rule_103: A21 -> 1
rule_104: A21 -> 1
rule_105: A21 -> 1
rule_106: A21 -> 1
rule_107: A21 -> 1
rule_108: A21 -> A20
rule_109: A22 -> 1
rule_110: A23 -> 1
rule_111: A24 -> 1
rule_112: A25 -> 1
rule_113: A26 -> A6 A28
rule_114: A26 -> A19 A27
rule_115: A26 -> A24 A29
rule_116: A26 -> A30 A29
rule_117: A27 -> 1
rule_118: A27 -> 18
rule_119: A28 -> 1
rule_120: A29 -> 1
rule_121: A30 -> 1
rule_122: A31 -> A13 A32
rule_123: A32 -> 1
rule_124: A32 -> 20
rule_125: A33 -> A6 A35
rule_126: A33 -> A19 A34
rule_127: A34 -> 1
rule_128: A34 -> 22
rule_129: A35 -> 1
rule_130: A36 -> A19 A37 23
rule_131: A36 -> A24 A38
rule_132: A37 -> 1
rule_133: A37 -> 24
rule_134: A38 -> 1
rule_135: A39 -> A16 A41
rule_136: A39 -> A19 A40 25
rule_137: A39 -> A24 A42
rule_138: A40 -> 1
rule_139: A40 -> 26
rule_140: A41 -> 1
rule_141: A42 -> 1
rule_142: A43 -> A13 A44
rule_143: A44 -> 1
rule_144: A44 -> 28
rule_145: A45 -> A13 A46 29
rule_146: A46 -> 1
rule_147: A46 -> 30
rule_148: A47 -> A3 A48
rule_149: A47 -> A6 A49 31
rule_150: A48 -> 1
rule_151: A49 -> 1
rule_152: A49 -> 32
rule_153: A50 -> A13 A51 33
rule_154: A50 -> A16 A51
rule_155: A51 -> 1
rule_156: A51 -> 34
rule_157: A52 -> A6 A54
rule_158: A52 -> A19 A53
rule_159: A52 -> A25 A55
rule_160: A53 -> 1
rule_161: A53 -> 36
rule_162: A54 -> 1
rule_163: A55 -> 1

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