Vector Algebra: JEE Main 2025 Mathematics question

Let the position vectors of three vertices of a triangle be $4 \vec{p} + \vec{q} - 3 \vec{r}$, $-5 \vec{p} + \vec{q} + 2 \vec{r}$, and $2 \vec{p} - \vec{q} + 2 \vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p} + \vec{q} + \vec{r}}{2}$ and $\alpha \vec{p} + \beta \vec{q} + \gamma \vec{r}$ respectively, then $\alpha + 2 \beta + 5 \gamma$ is equal to: