Lattice points on the trace 2

Given a square $ABCD$ with side length $a$. A curve is traced by point $D$ of the square as point $A$ slides along the x-axis and point $B$ slides along the y-axis.

The curve with $a=5$ is shown below.



For positive integer $a$, it can be shown that at least 4 lattice points are on the curve. An inscribed polygon can be formed by the lattice points on the curve. Let $R(a)$ be the ratio of the area of the inscribed polygon to the area of the closed curve with parameter $a$. Then $R(5)=0.942197263104$.

For $a\le 100$, the maximum value of $R(a)$ is $0.992825486421$, with $a=65$.

Find the maximum value of $R(a)$ and corresponding value $a$ for $a\le {10}^{15}$. If there are two or more values of $a$ taking the maximum value $R$, give the smallest $a$.

Answer format: [value of a],[value of R(a) (rounded to 12 digits after the decimal point)]

Example: 65,0.992825486421 for $a\le 100$.