Integral inequality $\int_{0}^{1}f^2\left(x\right)dx\geq\frac{3}{13}$

Question

Let $f: \left[0,\infty\right)$ $\rightarrow\left[0,\infty\right)$ be a continuous function so that $f(f(x))=x^2, \forall x\in \left[0,\infty\right).$ Show that: $$\int_{0}^{1}f^2\left(x\right)dx\geq\frac{3}{13} $$ I noticed $f$ is injective, so it is strictly monotone. Also, $f^2(x)=f(x^2), \forall x \geq 0$, but by substituting $x^2$ I'm getting nowhere. Do you have any ideas?