Let a =- i +2 j +2 k, b =8 i +7 j -3 k and c be a vector such that a × c = b. If c ·( i + j + k )=4, then | a + c |^2 is equal to:

Let $\displaystyle \overrightarrow{\mathrm{a}}=-\hat{i}+2 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=8 \hat{i}+7 \hat{j}-3 \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ be a vector such that $\displaystyle \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$. If $\displaystyle \overrightarrow{\mathrm{c}} \cdot(\hat{i}+\hat{j}+\hat{k})=4$, then $\displaystyle |\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}|^2$ is equal to :