Suppose that a function $f:[0, 1] \to (0, \infty)$ satisfies that
$$\int_0^1 f(x) dx = 1.$$
Prove the following inequality.
$$\left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx.$$

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