935. Rolling Square

A square of side length $b<1$$b < 1$ is rolling around the inside of a larger square of side length $1$$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=\frac{5}{13}$$b = \frac5{13}$.

For some values of $b$$b$, the small square may return to its initial position after several steps. For example, when $b=\frac{1}{2}$$b = \frac12$, this happens in $4$$4$ steps; and for $b=\frac{5}{13}$$b = \frac5{13}$ it happens in $24$$24$ steps.

Let $F(N)$$F(N)$ be the number of different values of $b$$b$ for which the small square first returns to its initial position within at most $N$$N$ steps. For example, $F(6)=4$$F(6) = 4$, with the corresponding $b$$b$ values:

the first three in $4$$4$ steps and the last one in $6$$6$ steps. Note that it does not matter whether the small square returns to its original orientation.
Also $F(100)=805$$F(100) = 805$.

Find $F({10}^8)$$F(10^8)$.

935. 滚动正方形

现有一边长为 $b<1$$b < 1$ 的小正方形在一个边长为 $1$$1$ 的大正方形内进行无滑动的滚动。初始时两个正方形共用一个内角。接下来的每一步中，小正方形会以某个在大正方形上的顶点为旋转中心，进行顺时针旋转，直至另一个顶点碰到大正方形。

下图展示了 $b=\frac{5}{13}$$b = \frac{5}{13}$ 时的前三步旋转：

对于某些 $b$$b$ 值，小正方形可能会在若干次旋转后回到初始位置。例如，$b=\frac{1}{2}$$b = \frac{1}{2}$ 时，小正方形会在 $4$$4$ 次旋转后回到原位。而 $b=\frac{5}{13}$$b = \frac{5}{13}$ 时则需要 $24$$24$ 次旋转。

记 $F(N)$$F(N)$ 为：能让小正方形在 $\le N$$\leq N$ 次旋转后回到原位的 $b$$b$ 的取值数。例如 $F(6)=4$$F(6) = 4$。因为 $b=\frac{1}{2},2-\sqrt{2},2+\sqrt{2}-\sqrt{2+4\sqrt{2}}$$b = \frac12, 2 - \sqrt 2, 2 + \sqrt 2 - \sqrt{2 + 4\sqrt2}$ 时，小正方形会在 $4$$4$ 次旋转后回到原位。$b=8-5\sqrt{2}+4\sqrt{3}-3\sqrt{6}$$b = 8 - 5\sqrt2 + 4\sqrt3 - 3\sqrt6$ 时，小正方形会在 $6$$6$ 次旋转后回到原位。除此之外没有其他情况。注意：小正方形返回原位时，其朝向不必和初始时的朝向相同。

亦已知 $F(100)=805$$F(100) = 805$。

求 $F({10}^8)$$F(10^8)$。

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