I'm trying to solve the following problem related to the mathematical explanation of Grover's algorithm. Let $$ \lvert\psi\rangle = \dfrac{1}{\sqrt{N}} \sum_{x=0}^{N-1}{\lvert x \rangle} \,\text{,} $$ then prove that the operator $HU_{0^{\perp}}H$ can be expressed as $\left( 2\lvert\psi\rangle\langle\psi\rvert-I \right)$, where the operator $U_{0^\perp}$ is defined as $$ U_{0^\perp}:\lvert x\rangle\mapsto-\lvert x\rangle \quad \forall\lvert x\rangle \ne \lvert 00...0 \rangle $$
However, I'm not able to see which steps should I follow. This problem is from the book An Introduction to Quantum Computing by Phillip Kaye, but it does not give any hint to solve it. I hope anyone can at least tell me where should I begin to get the answer.