HMMT 2023 U29

Let $P_{1}(x), P_{2}(x), \ldots, P_{k}(x)$ be monic polynomials of degree 13 with integer coefficients. Suppose there are pairwise distinct positive integers $n_{1}, n_{2}, \ldots, n_{k}$ for which, for all positive integers $i$ and $j$ less than or equal to $k$, the statement " $n_{i}$ divides $P_{j}(m)$ for every integer $m$ " holds if and only if $i=j$. Compute the largest possible value of $k$.