Let a =2 i - j +3 k, b =3 i -5 j + k and c be a vector such that a × c = c × b and ( a + c ) ·( b + c )=168. Then the maximum value of | c |^2 is:

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