Vector Algebra: JEE Main 2020 Mathematics question

Let the vectors $\overrightarrow a$, $\overrightarrow b$, $\overrightarrow c$ be such that $\left| {\overrightarrow a } \right| = 2$, $\left| {\overrightarrow b } \right| = 4$ and $\left| {\overrightarrow c } \right| = 4$. If the projection of $\overrightarrow b$ on $\overrightarrow a$ is equal to the projection of $\overrightarrow c$ on $\overrightarrow a$ and $\overrightarrow b$ is perpendicular to $\overrightarrow c$, then the value of $\left| {\overrightarrow a + \vec b - \overrightarrow c } \right|$ is ___________.