Let a complex number $z,\left|z\right|\ne 1,$ satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{\left|z\right|+11}{{(\left|z\right|-1)}^2}\right)\le 2.$ Then, the largest value of $\left|z\right|$is equal to

${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{\left|z\right|+11}{\left(\right|z|-1)}\right)\le 2$

$\Rightarrow \frac{\left|z\right|+11}{\left(\right|z|-1{)}^2}\ge \frac{1}{2}$

$\Rightarrow 2\left|z\right|+22\ge \left(\right|z|-1{)}^2$

$\Rightarrow 2\left|z\right|+22\ge |z{|}^2+1-2|z|$

$\Rightarrow |z{|}^2-4|z|-21\le 0$

$\Rightarrow \left(\right|z|-7)\left(\right|z|+3)\le 0$

$\Rightarrow \left|z\right|\in [0,7]$

So,  the largest value of  $\left|z\right|$ is 7