Vector Algebra: JEE Main 2026 Mathematics question

Let $\vec a=2\hat i-\hat j+\hat k$ and $\vec b=\lambda\hat j+2\hat k$, $\lambda\in\mathbb{Z}$ be two vectors. Let $\vec c=\vec a\times\vec b$ and $\vec d$ be a vector of magnitude $2$ in $yz$-plane. If $|\vec c|=\sqrt{53}$, then the maximum possible value of $(\vec c\cdot\vec d)^2$ is equal to :