Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $\left|\operatorname{Re}(\overline{z+2})\right|$ is:
\( \frac{2\sqrt{6}}{5} \)
\( \frac{24}{5} \)
\( \frac{1+\sqrt{6}}{5} \)
\( \frac{\sqrt{6}}{5} \)
