Consider two circles \( C_{1}: x^{2}+y^{2}=25 \) and \( C_{2}:(x-\alpha)^{2}+y^{2}=16 \), where \( \alpha \in (5,9) \). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \( C_{1} \) and \( C_{2} \) be \( \sin^{-1}\!\left(\dfrac{\sqrt{63}}{8}\right) \). If the length of common chord of \( C_{1} \) and \( C_{2} \) is \( \beta \), then the value of \( (\alpha \beta^{2}) \) equals ______.
