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Nishan Madushka and Kusal Mendis both made double centuries against Ireland Thursday as Sri Lanka's top four batters all hit hundreds for only the third time in Test history.
The hosts declared their first innings on 704 for three after tea on the fourth day of the second Test in Galle, 212 runs ahead.
Ireland were 54 for two at stumps, needing a further 158 on the final day to avoid a second straight innings defeat.
Ireland could still achieve an historic first draw if they can hang on after losing all five of their games since obtaining Test status in 2017.
Madushka converted his maiden Test century into a double hundred, becoming the second-youngest Sri Lankan to score a Test 200 after Mahela Jayawardene.
He was dismissed for 205 in the third over after lunch, lbw to Andy McBrine, with Ireland successfully reviewing the decision after the initial appeal was turned down.
Madushka's marathon knock lasted six minutes short of eight hours after he faced 339 deliveries and hit 22 fours and a six.
Mendis, who had twice been dismissed in the 190s, reached his first Test 200 by pushing a Ben White delivery behind square to steal a single.
He amassed 245 runs and was one short of Wasim Akram's world record of 12 sixes in a Test innings when he was caught at long-off attempting to equal that mark.
Angelo Mathews was dismissed for a duck in the first Test and was put down on one this match, but made the most of his reprieve to post his 16th Test hundred.
That gave the hosts their quadruple-centurions entry in the history books, after India against Bangladesh in 2007 and Pakistan against Sri Lanka in 2019.
Captain Dimuth Karunaratne, who scored 115 on Wednesday, declared immediately.
Their monster total of 704 for three was also the highest-ever score in Galle, eclipsing Bangladesh's 638 all out in 2013.
The pitch was still a batter's paradise but Ramesh Mendis cleaned up James McCollum for 10 before Jayasuriya had Peter Moor caught at short extra cover.
Andy Balbirnie and Harry Tector took Ireland to stumps without further trouble, with the captain on 12 not out and the Ireland number four on seven.