Special Pythagorean Triplet

A Pythagorean triplet is a set of three natural numbers, $a < b < c$ , for which, $a^{2} + b^{2} = c^{2} .$

For example, $3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}$ .

There exists exactly one Pythagorean triplet for which $a + b + c = 1000$ .
Find the product $a b c$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.