Erdős problem 202

Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$). How large can $r$ be in terms of $N$?

Let $f(N)$ be the maximum possible $r$, and let $L(N)=\exp(\sqrt{\log N\log\log N})$.

This was proved by GPT-5.4 Pro (prompted by Ho Boon Suan), using the argument of [BFV13] together with the resolution of the Kahn-Kalai conjecture by Park and Pham [PaPh24], so that $$f(N)= N L(N)^{-1+o(1)}.$$