Uniform, Bernoulli, and arcsine distributed random variables

$$X,Z,$$ and $$U$$ are independent random variables. $$X$$ is arcsine distributed random variable with $$\frac{1}{\pi\sqrt{x(1-x)}}$$ pdf on $$[0,1]$$ ($$0$$ otherwise), $$Z$$ is a Bernoulli distributed random variable with $$p=1/2,$$ and $$U$$ is a uniform distributed random variable on $$[0,1].$$ How can I prove, that $$UX+Z(1-X)$$ has the same distribution as $$X?$$

Take a bounded function $$f.$$ So we have $$E[f(UX+Z(1-X))]=\int_{0}^{1}\frac{1}{2\pi\sqrt{x(1-x)}}(\int_{0}^1(f(ux)+f(ux+1-x))du)dx=\int_0^1\frac{1}{2x\pi\sqrt{x(1-x)}}(\int_0^xf(y)dy+\int_{1-x}^1f(y)dy)dx=\int_{0}^1f(y)(\int_y^1\frac{1}{2\pi x\sqrt{x(1-x)}}dx+\int_{1-y}^1\frac{1}{2\pi x\sqrt{x(1-x)}}dx)dy=\int_0^1f(y)\frac{1}{\pi\sqrt{y(1-y)}}(1-y+y)dy=\int_0^1f(y)\frac{1}{\pi\sqrt{y(1-y)}}dy.$$