Let $\vec a = \hat i + \hat j + \hat k,\vec b = 3\hat i + 2\hat j - \hat k,\vec c = \lambda \hat j + \mu \hat k$ and $\hat d$ be a unit vector such that $\vec a \times \hat d = \vec b \times \hat d$ and $\vec c \cdot \hat d = 1$. If $\vec c$ is perpendicular to $\vec a$, then $|3\lambda \hat d + \mu \vec c{|^2}$ is equal to ______.