Mathematics Question from JEE Main 2025 · 4 April, Shift 2

Let the three sides of a triangle ABC be given by the vectors $2\hat{i} - \hat{j} + \hat{k}, \quad \hat{i} - 3\hat{j} - 5\hat{k}, \quad \text{and} \quad 3\hat{i} - 4\hat{j} - 4\hat{k}.$ Let G be the centroid of the triangle ABC. Then $6 \left( |\vec{AG}|^2 + |\vec{BG}|^2 + |\vec{CG}|^2 \right)$ is equal to _____