Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \dfrac{\text{length of arc }AB}{\text{length of arc }BC}=\dfrac{1}{5} \), and \( \overrightarrow{OC}=\alpha\,\overrightarrow{OA}+\beta\,\overrightarrow{OB} \), then \( a+\sqrt{2}(\sqrt{3}-1)\beta \) is equal to:
