7. Given the set $M=\left\{(a, b) \mid\left(y^{2}+4\right) a^{2}-\right.$ $2(x y+b y+8) a+x^{2}+2 b x+2 b^{2}+12$ is a square of a linear expression in $x$ and $y$ $\}$. When $(a, b)$ takes all elements in the set $M$, the maximum distance from the point $(a, b)$ to the origin $O$ is $\qquad$
$7 . \frac{2 \sqrt{21}}{3}$