Coordinate Geometry: JEE Main 2025 Mathematics question

Let $A$, $B$, $C$ be three points in $xy$-plane, whose position vectors are given by $\sqrt{3} \hat{i}+\hat{j}$, $\hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\vec{OA}$ and $\vec{OB}$ is $\frac{\theta}{\sqrt{2}}$, then the sum of all the possible values of $a$ is: