Let the position vectors of three vertices of a triangle be \(4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}\)…

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 \(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) and \(\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}\) respectively, then \(\alpha+2 \beta+5 \gamma\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]