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 21 terminals, ranging from 1 to 21.
Each terminal symbol occurs in the target sequence at least once.

There are 46 non-terminal symbols, ranging from A1 to A46.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> A38 A3
rule_3: A2 -> A38 A3
rule_4: A2 -> A38 A3
rule_5: A2 -> A38 A3
rule_6: A2 -> A38 A3
rule_7: A2 -> A38 A3
rule_8: A2 -> A38 A3
rule_9: A2 -> A38 A3
rule_10: A2 -> A38 A3
rule_11: A2 -> A38 A3
rule_12: A2 -> A38 A3
rule_13: A2 -> A38 A3
rule_14: A2 -> A38 A3
rule_15: A2 -> A38 A3
rule_16: A2 -> A38 A3
rule_17: A2 -> A38 A3
rule_18: A2 -> A38 A3
rule_19: A2 -> A38 A3
rule_20: A2 -> A38 A3
rule_21: A2 -> A38 A3
rule_22: A2 -> A38 A3
rule_23: A2 -> A38 A3
rule_24: A2 -> A38 A3
rule_25: A2 -> A38 A3
rule_26: A2 -> A38 A3
rule_27: A2 -> A38 A3
rule_28: A2 -> A38 A3
rule_29: A2 -> A38 A3
rule_30: A2 -> A38 A3
rule_31: A2 -> A38 A3
rule_32: A2 -> A38 A3
rule_33: A2 -> A38 A3
rule_34: A2 -> A38 A3
rule_35: A2 -> A38 A3
rule_36: A2 -> A38 A3
rule_37: A2 -> A38 A3
rule_38: A3 -> A4
rule_39: A3 -> A18
rule_40: A3 -> A28
rule_41: A4 -> A5
rule_42: A4 -> A5
rule_43: A4 -> A5
rule_44: A4 -> A5
rule_45: A4 -> A5
rule_46: A4 -> A5
rule_47: A4 -> A5
rule_48: A4 -> A5
rule_49: A4 -> A5
rule_50: A4 -> A5
rule_51: A4 -> A5
rule_52: A4 -> A5
rule_53: A4 -> A5
rule_54: A4 -> A5
rule_55: A4 -> A5
rule_56: A4 -> A5
rule_57: A4 -> A5
rule_58: A4 -> A5
rule_59: A4 -> A5
rule_60: A4 -> A5
rule_61: A4 -> A5
rule_62: A4 -> A5
rule_63: A4 -> A5
rule_64: A4 -> A5
rule_65: A4 -> A5
rule_66: A4 -> A5
rule_67: A4 -> A5
rule_68: A4 -> A5
rule_69: A4 -> A5
rule_70: A4 -> A5
rule_71: A4 -> A5
rule_72: A4 -> A5
rule_73: A4 -> A5
rule_74: A4 -> A5
rule_75: A4 -> A5
rule_76: A4 -> A5
rule_77: A4 -> A12
rule_78: A4 -> A12
rule_79: A4 -> A12
rule_80: A4 -> A12
rule_81: A4 -> A12
rule_82: A4 -> A12
rule_83: A4 -> A12
rule_84: A4 -> A12
rule_85: A4 -> A12
rule_86: A4 -> A12
rule_87: A4 -> A12
rule_88: A4 -> A12
rule_89: A4 -> A12
rule_90: A4 -> A12
rule_91: A4 -> A12
rule_92: A4 -> A12
rule_93: A4 -> A12
rule_94: A4 -> A12
rule_95: A4 -> A12
rule_96: A4 -> A12
rule_97: A4 -> A12
rule_98: A4 -> A12
rule_99: A4 -> A12
rule_100: A4 -> A12
rule_101: A4 -> A12
rule_102: A4 -> A12
rule_103: A4 -> A12
rule_104: A4 -> A12
rule_105: A4 -> A12
rule_106: A4 -> A12
rule_107: A4 -> A12
rule_108: A4 -> A12
rule_109: A4 -> A12
rule_110: A4 -> A12
rule_111: A4 -> A12
rule_112: A4 -> A12
rule_113: A4 -> A13
rule_114: A4 -> A13
rule_115: A4 -> A13
rule_116: A4 -> A13
rule_117: A4 -> A13
rule_118: A4 -> A13
rule_119: A4 -> A13
rule_120: A4 -> A13
rule_121: A4 -> A13
rule_122: A4 -> A13
rule_123: A4 -> A13
rule_124: A4 -> A13
rule_125: A4 -> A13
rule_126: A4 -> A13
rule_127: A4 -> A13
rule_128: A4 -> A13
rule_129: A4 -> A13
rule_130: A4 -> A13
rule_131: A4 -> A13
rule_132: A4 -> A13
rule_133: A4 -> A13
rule_134: A4 -> A13
rule_135: A4 -> A13
rule_136: A4 -> A13
rule_137: A4 -> A13
rule_138: A4 -> A13
rule_139: A4 -> A13
rule_140: A4 -> A13
rule_141: A4 -> A13
rule_142: A4 -> A13
rule_143: A4 -> A13
rule_144: A4 -> A13
rule_145: A4 -> A13
rule_146: A4 -> A13
rule_147: A4 -> A13
rule_148: A4 -> A13
rule_149: A5 -> 1
rule_150: A5 -> A5 A12
rule_151: A5 -> A6 A5
rule_152: A5 -> A9 A13
rule_153: A6 -> 1
rule_154: A6 -> A5 A11
rule_155: A6 -> A6 A6
rule_156: A6 -> A9 A7
rule_157: A7 -> 1
rule_158: A7 -> A7 A6
rule_159: A7 -> A8 A7
rule_160: A7 -> A13 A11
rule_161: A8 -> 1
rule_162: A8 -> A7 A9
rule_163: A8 -> A8 A8
rule_164: A8 -> A13 A10
rule_165: A9 -> 1
rule_166: A9 -> A5 A10
rule_167: A9 -> A6 A9
rule_168: A9 -> A9 A8
rule_169: A9 -> 21 13 A16 12 20 18 14
rule_170: A10 -> 1
rule_171: A10 -> A10 A8
rule_172: A10 -> A11 A9
rule_173: A10 -> A12 A10
rule_174: A11 -> 1
rule_175: A11 -> A10 A7
rule_176: A11 -> A11 A6
rule_177: A11 -> A12 A11
rule_178: A12 -> 1
rule_179: A12 -> A10 A13
rule_180: A12 -> A11 A5
rule_181: A12 -> A12 A12
rule_182: A13 -> 1
rule_183: A13 -> A7 A5
rule_184: A13 -> A8 A13
rule_185: A13 -> 11 5 8 9 A14 6 10
rule_186: A13 -> 11 8 A14 6 7
rule_187: A13 -> 11 8 A14 6 10 7
rule_188: A13 -> A13 A12
rule_189: A14 -> 1
rule_190: A14 -> A15 A14
rule_191: A15 -> 2
rule_192: A15 -> 3
rule_193: A15 -> 4
rule_194: A16 -> 1
rule_195: A16 -> A17 A16
rule_196: A17 -> 15
rule_197: A17 -> 16
rule_198: A17 -> 17
rule_199: A18 -> A19
rule_200: A18 -> A19
rule_201: A18 -> A19
rule_202: A18 -> A19
rule_203: A18 -> A19
rule_204: A18 -> A19
rule_205: A18 -> A19
rule_206: A18 -> A19
rule_207: A18 -> A19
rule_208: A18 -> A19
rule_209: A18 -> A19
rule_210: A18 -> A19
rule_211: A18 -> A19
rule_212: A18 -> A19
rule_213: A18 -> A19
rule_214: A18 -> A19
rule_215: A18 -> A19
rule_216: A18 -> A19
rule_217: A18 -> A19
rule_218: A18 -> A19
rule_219: A18 -> A19
rule_220: A18 -> A19
rule_221: A18 -> A19
rule_222: A18 -> A19
rule_223: A18 -> A19
rule_224: A18 -> A19
rule_225: A18 -> A19
rule_226: A18 -> A19
rule_227: A18 -> A19
rule_228: A18 -> A19
rule_229: A18 -> A19
rule_230: A18 -> A19
rule_231: A18 -> A19
rule_232: A18 -> A19
rule_233: A18 -> A19
rule_234: A18 -> A19
rule_235: A18 -> A26
rule_236: A18 -> A26
rule_237: A18 -> A26
rule_238: A18 -> A26
rule_239: A18 -> A26
rule_240: A18 -> A26
rule_241: A18 -> A26
rule_242: A18 -> A26
rule_243: A18 -> A26
rule_244: A18 -> A26
rule_245: A18 -> A26
rule_246: A18 -> A26
rule_247: A18 -> A26
rule_248: A18 -> A26
rule_249: A18 -> A26
rule_250: A18 -> A26
rule_251: A18 -> A26
rule_252: A18 -> A26
rule_253: A18 -> A26
rule_254: A18 -> A26
rule_255: A18 -> A26
rule_256: A18 -> A26
rule_257: A18 -> A26
rule_258: A18 -> A26
rule_259: A18 -> A26
rule_260: A18 -> A26
rule_261: A18 -> A26
rule_262: A18 -> A26
rule_263: A18 -> A26
rule_264: A18 -> A26
rule_265: A18 -> A26
rule_266: A18 -> A26
rule_267: A18 -> A26
rule_268: A18 -> A26
rule_269: A18 -> A26
rule_270: A18 -> A26
rule_271: A18 -> A27
rule_272: A18 -> A27
rule_273: A18 -> A27
rule_274: A18 -> A27
rule_275: A18 -> A27
rule_276: A18 -> A27
rule_277: A18 -> A27
rule_278: A18 -> A27
rule_279: A18 -> A27
rule_280: A18 -> A27
rule_281: A18 -> A27
rule_282: A18 -> A27
rule_283: A18 -> A27
rule_284: A18 -> A27
rule_285: A18 -> A27
rule_286: A18 -> A27
rule_287: A18 -> A27
rule_288: A18 -> A27
rule_289: A18 -> A27
rule_290: A18 -> A27
rule_291: A18 -> A27
rule_292: A18 -> A27
rule_293: A18 -> A27
rule_294: A18 -> A27
rule_295: A18 -> A27
rule_296: A18 -> A27
rule_297: A18 -> A27
rule_298: A18 -> A27
rule_299: A18 -> A27
rule_300: A18 -> A27
rule_301: A18 -> A27
rule_302: A18 -> A27
rule_303: A18 -> A27
rule_304: A18 -> A27
rule_305: A18 -> A27
rule_306: A18 -> A27
rule_307: A19 -> 1
rule_308: A19 -> A19 A26
rule_309: A19 -> A20 A19
rule_310: A19 -> A23 A27
rule_311: A20 -> 1
rule_312: A20 -> A19 A25
rule_313: A20 -> A20 A20
rule_314: A20 -> A23 A21
rule_315: A21 -> 1
rule_316: A21 -> A21 A20
rule_317: A21 -> A22 A21
rule_318: A21 -> A27 A25
rule_319: A22 -> 1
rule_320: A22 -> A21 A23
rule_321: A22 -> A22 A22
rule_322: A22 -> A27 A24
rule_323: A23 -> 1
rule_324: A23 -> A19 A24
rule_325: A23 -> A20 A23
rule_326: A23 -> 21 14 13 A16 12 18
rule_327: A23 -> A23 A22
rule_328: A24 -> 1
rule_329: A24 -> A24 A22
rule_330: A24 -> A25 A23
rule_331: A24 -> A26 A24
rule_332: A25 -> 1
rule_333: A25 -> A24 A21
rule_334: A25 -> A25 A20
rule_335: A25 -> A26 A25
rule_336: A26 -> 1
rule_337: A26 -> A24 A27
rule_338: A26 -> A25 A19
rule_339: A26 -> A26 A26
rule_340: A27 -> 1
rule_341: A27 -> 11 5 8 A14 6 10 7 5
rule_342: A27 -> A21 A19
rule_343: A27 -> A22 A27
rule_344: A27 -> A27 A26
rule_345: A28 -> A29
rule_346: A28 -> A29
rule_347: A28 -> A29
rule_348: A28 -> A29
rule_349: A28 -> A29
rule_350: A28 -> A29
rule_351: A28 -> A29
rule_352: A28 -> A29
rule_353: A28 -> A29
rule_354: A28 -> A29
rule_355: A28 -> A29
rule_356: A28 -> A29
rule_357: A28 -> A29
rule_358: A28 -> A29
rule_359: A28 -> A29
rule_360: A28 -> A29
rule_361: A28 -> A29
rule_362: A28 -> A29
rule_363: A28 -> A29
rule_364: A28 -> A29
rule_365: A28 -> A29
rule_366: A28 -> A29
rule_367: A28 -> A29
rule_368: A28 -> A29
rule_369: A28 -> A29
rule_370: A28 -> A29
rule_371: A28 -> A29
rule_372: A28 -> A29
rule_373: A28 -> A29
rule_374: A28 -> A29
rule_375: A28 -> A29
rule_376: A28 -> A29
rule_377: A28 -> A29
rule_378: A28 -> A29
rule_379: A28 -> A29
rule_380: A28 -> A29
rule_381: A28 -> A36
rule_382: A28 -> A36
rule_383: A28 -> A36
rule_384: A28 -> A36
rule_385: A28 -> A36
rule_386: A28 -> A36
rule_387: A28 -> A36
rule_388: A28 -> A36
rule_389: A28 -> A36
rule_390: A28 -> A36
rule_391: A28 -> A36
rule_392: A28 -> A36
rule_393: A28 -> A36
rule_394: A28 -> A36
rule_395: A28 -> A36
rule_396: A28 -> A36
rule_397: A28 -> A36
rule_398: A28 -> A36
rule_399: A28 -> A36
rule_400: A28 -> A36
rule_401: A28 -> A36
rule_402: A28 -> A36
rule_403: A28 -> A36
rule_404: A28 -> A36
rule_405: A28 -> A36
rule_406: A28 -> A36
rule_407: A28 -> A36
rule_408: A28 -> A36
rule_409: A28 -> A36
rule_410: A28 -> A36
rule_411: A28 -> A36
rule_412: A28 -> A36
rule_413: A28 -> A36
rule_414: A28 -> A36
rule_415: A28 -> A36
rule_416: A28 -> A36
rule_417: A28 -> A37
rule_418: A28 -> A37
rule_419: A28 -> A37
rule_420: A28 -> A37
rule_421: A28 -> A37
rule_422: A28 -> A37
rule_423: A28 -> A37
rule_424: A28 -> A37
rule_425: A28 -> A37
rule_426: A28 -> A37
rule_427: A28 -> A37
rule_428: A28 -> A37
rule_429: A28 -> A37
rule_430: A28 -> A37
rule_431: A28 -> A37
rule_432: A28 -> A37
rule_433: A28 -> A37
rule_434: A28 -> A37
rule_435: A28 -> A37
rule_436: A28 -> A37
rule_437: A28 -> A37
rule_438: A28 -> A37
rule_439: A28 -> A37
rule_440: A28 -> A37
rule_441: A28 -> A37
rule_442: A28 -> A37
rule_443: A28 -> A37
rule_444: A28 -> A37
rule_445: A28 -> A37
rule_446: A28 -> A37
rule_447: A28 -> A37
rule_448: A28 -> A37
rule_449: A28 -> A37
rule_450: A28 -> A37
rule_451: A28 -> A37
rule_452: A28 -> A37
rule_453: A29 -> 1
rule_454: A29 -> A29 A36
rule_455: A29 -> A30 A29
rule_456: A29 -> A33 A37
rule_457: A30 -> 1
rule_458: A30 -> A29 A35
rule_459: A30 -> A30 A30
rule_460: A30 -> A33 A31
rule_461: A31 -> 1
rule_462: A31 -> A31 A30
rule_463: A31 -> A32 A31
rule_464: A31 -> A37 A35
rule_465: A32 -> 1
rule_466: A32 -> A31 A33
rule_467: A32 -> A32 A32
rule_468: A32 -> A37 A34
rule_469: A33 -> 1
rule_470: A33 -> 21 14 A16 18
rule_471: A33 -> A29 A34
rule_472: A33 -> A30 A33
rule_473: A33 -> A33 A32
rule_474: A34 -> 1
rule_475: A34 -> A34 A32
rule_476: A34 -> A35 A33
rule_477: A34 -> A36 A34
rule_478: A35 -> 1
rule_479: A35 -> A34 A31
rule_480: A35 -> A35 A30
rule_481: A35 -> A36 A35
rule_482: A36 -> 1
rule_483: A36 -> A34 A37
rule_484: A36 -> A35 A29
rule_485: A36 -> A36 A36
rule_486: A37 -> 1
rule_487: A37 -> 11 5 9 A14 6 5
rule_488: A37 -> A31 A29
rule_489: A37 -> A32 A37
rule_490: A37 -> A37 A36
rule_491: A38 -> A40 A39
rule_492: A38 -> A43 A41
rule_493: A38 -> A46 A44
rule_494: A39 -> 1
rule_495: A39 -> 1
rule_496: A39 -> 1
rule_497: A39 -> 1
rule_498: A39 -> 1
rule_499: A39 -> 1
rule_500: A39 -> 1
rule_501: A39 -> 1
rule_502: A39 -> 1
rule_503: A39 -> 1
rule_504: A39 -> 1
rule_505: A39 -> 1
rule_506: A39 -> 1
rule_507: A39 -> 1
rule_508: A39 -> 1
rule_509: A39 -> 1
rule_510: A40 -> 1
rule_511: A40 -> 1
rule_512: A40 -> 1
rule_513: A41 -> 1
rule_514: A41 -> 1
rule_515: A41 -> 1
rule_516: A41 -> 1
rule_517: A41 -> 1
rule_518: A41 -> 1
rule_519: A41 -> 1
rule_520: A41 -> 1
rule_521: A41 -> 1
rule_522: A41 -> 1
rule_523: A41 -> 1
rule_524: A41 -> 1
rule_525: A41 -> 1
rule_526: A41 -> 1
rule_527: A41 -> 1
rule_528: A41 -> 1
rule_529: A41 -> A42 A41
rule_530: A42 -> 14
rule_531: A42 -> 18
rule_532: A43 -> 1
rule_533: A43 -> 1
rule_534: A43 -> 1
rule_535: A43 -> 12
rule_536: A43 -> 21
rule_537: A44 -> 1
rule_538: A44 -> 1
rule_539: A44 -> 1
rule_540: A44 -> 1
rule_541: A44 -> 1
rule_542: A44 -> 1
rule_543: A44 -> 1
rule_544: A44 -> 1
rule_545: A44 -> 1
rule_546: A44 -> 1
rule_547: A44 -> 1
rule_548: A44 -> 1
rule_549: A44 -> 1
rule_550: A44 -> 1
rule_551: A44 -> 1
rule_552: A44 -> 1
rule_553: A44 -> A45 A44
rule_554: A45 -> 5
rule_555: A45 -> 7
rule_556: A46 -> 1
rule_557: A46 -> 1
rule_558: A46 -> 1
rule_559: A46 -> 6
rule_560: A46 -> 11

Target sequence:
1 1 21 14 13 19 17 16 15 1 12 20 18 14 11 5 8 9 4 2 3 1 6 10 7 5 1
"""