number theory - Non-negative rational integer composed of primes may be written as the sum of two squares

Show that any non-negative rational integer of the form

-- where the $p_1,p_2,...,p_m$ are all primes congruent to 1 mod 4, the $q_1,q_2,...,q_n$ are all primes congruent to 3 mod 4 and $i,k_1,k_2,...,k_m,j_1,j_2,...,j_n$ are all non-negative integers --
may be written as the sum of two squares.

My Work So Far:

• I showed that if x and y may both be written as the sum of the squares of two integers then their product may also be written as the sum of two integers squared



• I showed that if p is an integer prime congruent to 1 mod 4 then p is not a prime in Q[$\sqrt{-1}$]


• I showed that if p is an integer prime congruent to 1 mod 4 then there are rational integers a and b for which $p=a^2+b^2$

I'm just not sure how to combine these results. Any help would be welcome, thank you!