IrMO 2019 Problem 1

Define the quasi-primes as follows. $\bullet$ The first quasi-prime is $q_1 = 2$ $\bullet$ For $n \ge 2$ , the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$ . Determine, with proof, whether or not $1000$ is a quasi-prime.