100 Prisoners and 100 Boxes

100 prisoners are numbered 1 to 100. In a room, 100 boxes each contain one slip with a distinct number 1 to 100, placed by a uniformly random permutation. Each prisoner enters alone, may open at most 50 boxes, must find the slip bearing his own number, then leaves without communicating or altering anything. All 100 must succeed for the group to win. Using the optimal strategy (each prisoner opens the box with his number, then the box whose number matches the slip just found, following the permutation cycle), the win probability equals 1 minus the sum of 1/k for k from 51 to 100. To the nearest whole percent, what is this winning probability?