Let $\vec a=\hat i+\hat j+\sqrt{2}\,\hat k$, $\vec b=b_1\hat i+b_2\hat j+\sqrt{2}\,\hat k$, $\vec c=5\hat i+\hat j+\sqrt{2}\,\hat k$ be three vectors such that the projection vector of $\vec b$ on $\vec a$ is $\vec a$. If $\vec a+\vec b$ is perpendicular to $\vec c$, then $|\vec b|$ is equal to: