Let \(\vec{a}=2\overset{^}{i}-\overset{^}{j}+3\overset{^}{k},\vec{b}=3\overset{^}{i}-5\overset{^}{j}+\overset{^}{k}\text…

Let \(\vec{a}=2\overset{^}{i}-\overset{^}{j}+3\overset{^}{k},\vec{b}=3\overset{^}{i}-5\overset{^}{j}+\overset{^}{k}\text{ and }\vec{c}\)

be a vector such that \(\vec{\mathrm{a}}\times \vec{\mathrm{c}}=\vec{\mathrm{c}}\times \vec{\mathrm{b}}\) and

\(\left(\vec{a}+\vec{c}\right)\cdot \left(\vec{b}+\vec{c}\right)=168\). Then the maximum

value of \(|\vec{\mathrm{c}}{|}^{2}\) is:

[JEE Main 2025, 29 Jan (Shift 1)]