Eccoci.

A square of side length $b<1$ is rolling around the inside of a larger square of side length $1$, always touching the larger square but without sliding.
Initially the two squares share a common corner. At each step, the small square rotates clockwise about a corner that touches the large square, until another of its corners touches the large square. Here is an illustration of the first three steps for $b = \frac5{13}$.

For some values of $b$, the small square may return to its initial position after several steps. For example, when $b = \frac12$, this happens in $4$ steps; and for $b = \frac5{13}$ it happens in $24$ steps.

Let $F(N)$ be the number of different values of $b$ for which the small square first returns to its initial position within at most $N$ steps. For example, $F(6) = 4$, with the corresponding $b$ values: $$\frac{1}{2},2-\sqrt{2},2+\sqrt{2}-\sqrt{2+4\sqrt{2}},8-5\sqrt{2}+4\sqrt{3}-3\sqrt{6},$$ the first three in $4$ steps and the last one in $6$ steps. Note that it does not matter whether the small square returns to its original orientation.
Also $F(100) = 805$.

Find $F(10^8)$.