Three Dimensional Geometry: JEE Main 2025 Mathematics question

If the equation of the line passing through the point $(0, -\tfrac{1}{2}, 0)$ and perpendicular to the lines $\mathbf{r} = \lambda(\hat{i} + a\hat{j} + b\hat{k})$ and $\mathbf{r} = (\hat{i} - \hat{j} - 6\hat{k}) + \mu(-b\hat{i} + a\hat{j} + 5\hat{k})$ is $\frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}$, then $a + b + c + d$ is equal to: