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 $abc$.
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