Algebra Level 5

Three real numbers $$x, y,$$ and $$z$$ such that $$x < y < z$$ satisfy the following three equations:

\begin{align} x + y + z &= 4 \\ x^2 + y^2 + z^2 &= 10 \\ x^3 + y^3 + z^3 &= 22. \end{align}

Find the value of $$z.$$

This problem is not original.

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