Vector Algebra: JEE Main 2021 Mathematics question

Let $\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$ and $\overrightarrow q = \widehat i + 2\widehat j + \widehat k$ be two vectors. If a vector $\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)$ is perpendicular to each of the vectors ($(\overrightarrow p + \overrightarrow q )$ and $(\overrightarrow p - \overrightarrow q )$, and $\left| {\overrightarrow r } \right| = \sqrt 3$, then $\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|$ is equal to _______________.