Two times, Three times, Thirsty and SAY WHEN???

Level pending

In a Cartesian plane, Let \( A \) be the set of all natural less than \(15\) such that there exists a triangle with side length \( n,n+1,n+2 \) which all vertices are lattice points.

Let \( B \) \( =\) {\( x^3\) |\(x \)\(\in\)\( A \) }.

Let \( s \) the sum of the elements of \( B \) . Find the first three digits of \( s \)

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