Proof Writing & Problem Solving · Superchallenge problem

Call a positive integer a Hilbert number if it leaves remainder 1 on division by 4 (so $1, 5, 9, 13, 17, 21, \ldots$); products of Hilbert numbers are Hilbert numbers. A Hilbert number greater than 1 is a Hilbert prime if it is not the product of two smaller Hilbert numbers greater than 1. Factorisation into Hilbert primes EXISTS but is not unique. In how many ways, ignoring order, can $53361 = 3^2 \cdot 7^2 \cdot 11^2$ be written as a product of Hilbert primes?