Triangles inside Circles

For a given integer $R$ consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with radius $R$ . Define $F (R)$ to be the largest inradius of those triangles. You are given $F (100) = 36$ .

Find $F (10^{18})$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.