Vector Algebra: JEE Main 2022 Mathematics question

Let $\overrightarrow a$ be a vector which is perpendicular to the vector $3\widehat i + {1 \over 2}\widehat j + 2\widehat k$. If $\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\widehat k$, then the projection of the vector $\overrightarrow a$ on the vector $2\widehat i + 2\widehat j + \widehat k$ is :