10 dimensional jump

Given a set of points in space, a jump consists of taking two points in the set, \(P\) and \(Q\), removing \(P\) from the set, and replacing it with the reflection of \(P\) over \(Q\). Find the smallest number \(n\) such that for any set of \(n\) lattice points in \(10\)-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.

This problem is from the OMO.

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