Let circle $$C$$ be the image of $$x^2 + y^2 - 2x + 4y - 4 = 0$$ in the line $$2x - 3y + 5 = 0$$ and $$A$$ be the point on $$C$$ such that $$OA$$ is parallel to the $$x$$-axis and $$A$$ lies on the right hand side of the centre $$O$$ of $$C$$. If $$B(\alpha, \beta)$$, with $$\beta < 4$$, lies on $$C$$ such that the length of the arc $$AB$$ is $$\left(1/6\right)^{\text{th}}$$ of the perimeter of $$C$$, then $$\beta + \sqrt{3}\alpha$$ is equal to