PL-OM 2017 Problem 3

Integers $a_1, a_2, \ldots, a_n$ satisfy $$1<a_1<a_2<\ldots < a_n < 2a_1.$$ If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$ , then prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$