Let \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \quad \vec{b}=3 \hat{i}-5 \hat{j}+\widehat{k}\) and \(\vec{c} \) be a vector …

Let \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \quad \vec{b}=3 \hat{i}-5 \hat{j}+\widehat{k}\) and \(\vec{c} \) be a vector such that \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}} \) and \((\vec{a}+\vec{c}) \cdot(\vec{b}+\vec{c})=168 \). Then the maximum value of \(|\overrightarrow{\mathrm{c}}|^2 \) is:

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