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balanced prime  and  plaindrome

First numbers that belong to both sets:

5, 157, 257, 1123, 1223, 1367, 2677, 4457, 11299, 13457, 22259, 22447, 23333, 24677, 36677, 46889, 122579, 124459, 145589, 224467, 233347, 244457, 256889, 289999, 334777, 335579, 344599, 555677, 566999, 688889, 1111169, 1111667, 1112477, 1114999, 1115567, 1117889, 1123777, 1222789, 1244459, 1244567, 1667899, 2227789, 2235677, 2245799, 2335679, 2555699, 2566667, 3335567, 3355577, 3444499, 5555777, 5688899, 11111899, 11234689, 11246699, 11333489, 11335567, 11335679, 11357777, 11455559, 12224669, 12356789, 13333589, 13336669, 13445567, 14444459, 14557889, 16678889, 22223899, 22344677, 23333467, 23366899, 23444999, 23446679, 24444599, 33344777, 33377779, 35555777, 35788889, 44456779, 45566779, 55567789, 111117779, 111334459, 111677999, 112346699, 112445777, 115556699, 115579999, 115777789, 122236799, 122267777, 122333579, 122455679, 122677799, 123445799, 123555569, 123566777, 125555567, 133445579, 133667777, 134448889, 144455999, 144577777, 146778899, 147778889, 222224677, 222234479, 222344477, 222346667, 222668899, 223346999, 223577789, 223677889, 224556679, 224778889, 225567889, 227788889, 234466889, 238888889, 244458899, 244566677, 256777777, 257777899, 333358999, 333488899, 335668999, 344447777, 344479999, 344557999, 346679999, 346999999, 355888889, 445556789, 455555689, 666667789, 779999999, 799999999, 1111112677, 1111266677, 1111336679, 1111444567, 1112244449, 1112333557, 1112334569, 1115678999, 1122224567, 1122234467, 1122336667, 1122355567, 1123556677, 1123677899, 1124455799, 1124555569, 1124567779, 1133455559, 1135558999, 1222344479, 1222388899, 1223334599, 1223344489, 1224445799, 1226677889, 1234444667, 1234456889, 1235555567, 1236667799, 1244456999, 1245667777, 1245777899, 1246666679, 1267799999, 1336789999, 1345799999, 1346888999, 1355888999, 1356788999, 1367778899, 1444445557, 1444445789, 1444666889, 1455677789, 1477778899, 1667888899, 2222222267, 2222268889, 2222345599, 2224448999, 2225556679, 2233333567, 2234446777, 2244455699, 2257778899, 2266788889, 2333336777, 2333356679, 2333447777, 2334446689, 2335567889, 2366678999, 2444444567, 2455556669, 2555566799, 2557777799, 3334445899, 3334455569, 3334555669, 3338888899, 3344445667, 3345555779, 3345689999, 3366677999, 3444466777, 3445888889, 3455666777, 4444556899, 4445555689, 5555577889, 5556666677, 6888899999.