
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 $\beta = bi$ where $b \neq 0$ and $b$ is a real number, and $\alpha$ be a complex number such that $\overline{\alpha} \beta \neq -1$. Find the maximum value of
\[
\left| \frac{\beta - \alpha}{1 + \overline{\alpha} \beta} \right|.
\]

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