Nemotron-IMO-Bench 2026 N24

Let $(a_n)_{n\geq 1}$ be a sequence of positive integers such that any prime larger than $1402$ divides some term of the sequence and let $b_n=a_1...a_n-1$. Prove that infinitely many primes divide at least one term of the sequence $(b_n)_{n\geq 1}$.