An Intersection on a Fixed Circle

Consider the hyperbola $\dfrac{x^2}2-y^2=1$ with fixed points $A(-2,1)$, $B(2,1)$ on it. A line through the fixed point $Q(0,-3)$ meets the hyperbola at points $C$ and $D$; the lines $AC$ and $BD$ meet at a point $P$. Prove that $P$ lies on a fixed circle.