Nemotron-IMO-Bench 2026 N46

Let $p$ be an odd prime not dividing $abcd$, where $a,b,c,d$ are pairwise distinct integers. Let $v_p(x)$ be the largest nonnegative integer $k$ such that $p^k$ divides the nonzero integer $x$. If the sequence $(v_p(ca^n-db^n))_{n\geq 1}$ takes at least one positive value and is bounded above, prove that it takes exactly one positive value.