Let \(\vec{a}=2\overset{^}{i}−\overset{^}{j}+\overset{^}{k}\) and \(\vec{b}=\lambda \overset{^}{j}+2\overset{^}{k},\text…

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

[JEE Main 2026, 22 Jan (Shift 2)]