Mathematics Question from JEE Main 2026 · 22 January, Shift 2

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