Go Factor!

\[\large 399\:337 = 616^2 + 141^2 \\ \large 399\:337 = 464^2 + 429^2\] The number \(399\:337\) is the product of two prime numbers \(p\times q\). Determine these prime numbers without using a calculator.

Submit your answer as the sum \(p + q\).

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