Black captures three white stones with 53 and saves the two black ones. Those three white stones have been dead for a long time, but until now Black has never had a reason to capture. White plays 54 to reduce Black's territory in the corner as much as possible.
Black makes a ko with 55, putting two white stones into atari at the same time. White saves them by playing 56. Now Black could play in the corner to end the ko, but this is not the right idea. With 57, Black gains one sure point on the right side, and still has a chance to win the ko later.
White takes the ko with 58 but Black plays 59 and says "atari". If White ends the first ko fight, Black will start a second one. White wisely plays 60 to make sure that the white stones in the top right corner stay alive.
Since the ko threat was answered, Black takes back the ko with 61. White
finds a nice threat with 62, which puts a black stone into atari and threatens
to take a big bite out of Black's territory, and 63 defends against this
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