Is there a way to obtain $A^+$ if we know $A$ and $P=AA^+$

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Suppose you could obtain $P:=AA^+\in\mathbb{R}^{N\times N}$ experimentally, where $A^+$ is the Moore-Penrose Pseudoinverse. Is there a way to obtain $A^+$?

We know that rank$(A)=N-1$.