Proof Writing & Problem Solving · Hard problem

A proof that there are infinitely many primes of the form $4k + 3$ takes such primes $p_1, \ldots, p_k$, forms $N = 4 p_1 p_2 \cdots p_k - 1$, and argues that $N$ has a prime factor of the form $4k + 3$ that is not on the list. Checking the argument with the list $3, 7, 11$, the writer finds $N = 923$. Which prime factor of $923$ has the form $4k + 3$?