**True or False**.

Let there be a complex number \(z\) such that

\[\large \begin{cases} |z| = 1 \\ |\Re(z)| \neq 1 \end{cases} \]

Then \( i \left( \dfrac{ z-1}{z+1} \right) \) is **always** a real number.

**Notations:**

- \( |\cdot| \) denotes the absolute value function.
- \( \Re(\cdot) \) denotes the real part function.

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