Ohh, squares

Two positive integers (\(x, y\)) satisfy that \(xy+1\) is a perfect square.

True or false?

There always exist a positive integer (\(z\)), where all of the numbers below are perfect squares. \[\begin{align} yz & + 1 \\ xz & + 1 \\ xy+xz+yz & + 1 \end{align}\]

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