The incenter of a triangle

ABC is an integer sided triangle with incenter I and perimeter p. The segments IA, IB and IC have integral length as well.

Let L = p + |IA| + |IB| + |IC|.

Let S(P) = ∑L for all such triangles where p ≤ P. For example, S(10³) = 3619.

Find S(10⁷).

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