Vector Algebra: JEE Main 2026 Mathematics question

Let $\vec a=2\hat i-\hat j-\hat k$, $\vec b=\hat i+3\hat j-\hat k$ and $\vec c=2\hat i+\hat j+3\hat k$. Let $\vec v$ be the vector in the plane of the vectors $\vec a$ and $\vec b$, such that the length of its projection on the vector $\vec c$ is $\frac{1}{\sqrt{14}}$. Then $|\vec v|$ is equal to