Eccoci.

A Pythagorean triangle is called supernatural if two of its three sides are consecutive integers.

Let $S(N)$ be the sum of the perimeters of all distinct supernatural triangles with perimeters less than or equal to $N$.
For example, $S(100) = 258$ and $S(10000) = 172004$.

Find $S(10^{10^{10} })$. Give your answer modulo $1234567891$.