= Solution
Condition on the fair coin, taken independent of $U$. For the <squared uniform distribution>, the <cumulative distribution function> of $U^2$ is $\sqrt{x}$ on $0<x<1$, giving <probability density function> $1/(2\sqrt{x})$. Reflection about $1/2$ gives the <probability density function> of $1-U^2$ as $1/(2\sqrt{1-x})$. Averaging the two conditional <probability density functions> produces this <mixture model>:
$$
\boxed{f_X(x)=\begin{cases}\dfrac1{4\sqrt{x}}+\dfrac1{4\sqrt{1-x}}&0<x<1,\\0&\text{otherwise}.\end{cases}}
$$
There are no atoms at the endpoints; each integrable singularity contributes half the total mass. Equivalently the <cumulative distribution function> is $[\sqrt{x}+1-\sqrt{1-x}]/2$ on the interval.
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