Largest Inscribed Quadrilateral in a Circular Board

A circular wooden board is measured with a carpenter's square, an L-shaped tool formed by two perpendicular straight edges. As shown, the square is placed against the board so that its right-angle corner and the two outer endpoints of its legs all lie on the circle; the legs have lengths $5\text{ cm}$ and $10\text{ cm}$.

A quadrilateral is then cut from the board so that all four of its vertices lie on the circle and one of its interior angles $\alpha$ satisfies $\cos\alpha=\dfrac13$. Find the maximum possible perimeter of this quadrilateral.