US-EGMO-TST 2026 Problem 4

For a positive integer $n$, let $c_n$ be the smallest possible value of $\max(a, b)$ over all pairs of distinct positive integers $a$ and $b$ such that

$$\operatorname{lcm}(a, b) + k = \operatorname{lcm}(a + k, b + k)$$

for all $0 \le k \le n$. Find all positive integers $n$ for which $c_n = c_{n+1}$.