Derive a UMP test

Question

Let $F$ and $G$ be two known absolutely continous c.d.f's on the real line with p.d.f's $f$ and $ g$. Assume we have a single observation $X$ drawn from the c.d.f: $$\theta F(x)+(1-\theta)G(x),~~\theta\in(0,1)$$ Given $0<\theta_{0}<1$, how to derive a UMP test of size $\alpha$ for testing: $$H_0:\theta\leq\theta_0~~vs.~~H_1:\theta>\theta_0$$ My thought was to find a LRT, or to use Karlin–Rubin theorem, but didn't see a way to work it out.