The solution to this problem is a two stage ko. If White cannot win this ko, then he can back out with a subsequently.
A connection at is highly undesirable for White. After
, the ko becomes a direct ko, and the White group is extremely heavy.
The atari at is a failure. Instead of capturing at a, White can descend at
and there is nothing else Black can do.
The atari at here does not work either. The key move for White is
, which prevents the ko.
The following discussion is pitched at a slightly higher level than the problem, but it is still revelant to the problem.
Without the black stone at a, can extend and say arigato to Black for the free gift. Obviously White should not play
at b otherwise black can play at
and thank White instead.
So without Black a, Black should atari at . The correct response is
, regardless of whether there is a Black stone at a. This is probably a IGS 1k* level indicator.