HMMT 2025 U32

In the coordinate plane, a closed lattice loop of length $2n$ is a sequence of lattice points $P_{0}, P_{1}, P_{2}, \ldots , P_{2n}$ such that $P_{0}$ and $P_{2n}$ are both the origin and $P_{i}P_{i + 1} = 1$ for each $i$ . A closed lattice loop of length 2026 is chosen uniformly at random from all such loops. Let $k$ be the maximum integer such that the line $\ell$ with equation $x + y = k$ passes through at least one point of the loop. Compute the expected number of indices $i$ such that $0 \leq i \leq 2025$ and $P_{i}$ lies on $\ell$ .

(A lattice point is a point with integer coordinates.)