
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $\alpha$ and $\beta$ be complex numbers such that $|\beta| = 1$ and $\overline{\alpha} \beta \neq \omega$, where $\omega$ is a non-zero complex number with $|\omega| <1$. Find the maximum value of
\[
\left| \frac{\omega\beta - \alpha}{1 - \overline{\alpha} \beta} \right|.
\]

Present the answer in LaTex format: \boxed{Your answer}
user