Special Pythagorean triplet

A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, 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 $abc$.