Given a circle (let's call it circle \(O\) ) inscribed in a triangle \(XYZ\) with \( XY \neq XZ\). The circle \(O\) touches \(YZ\), \(ZX\), and \(XY\) at \(U\), \(V\), and \(W\) respectively. Point \(R\) lies on \(XZ\) and point \(S\) lies on \(XY\), such that \(RS\) and \(YZ\) are parallel to each other. Let \(P\) be a circle that passes through the point \(R\) and \(S\), and \(P\) touches the circle \(O\) at \(T\). Prove that \(VW\), \(UT\), and \(RS\) intersect at one point.