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 \( abc \).