100 Prisoners and 100 Boxes

100 prisoners are numbered $1$ through $100$. A room contains 100 boxes, also numbered $1$–$100$. Each box contains a slip with one prisoner's number, placed in a uniformly random permutation.

One by one, prisoners enter and may open up to 50 boxes. To survive, each prisoner must find the slip with their own number. They can plan a strategy beforehand but cannot communicate once it starts.

Maximize the probability that all 100 prisoners survive.