A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)$, where $a$ and $b$ are positive integers. What is $a+b$? $\textbf{(A)} ~64 \qquad$ $\textbf{(B)} ~66 \qquad$ $\textbf{(C)} ~68 \qquad$ $\textbf{(D)} ~70 \qquad$ $\textbf{(E)} ~72$