
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
In a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), one of the acute angles \( \beta \) satisfies
\[
\tan \frac{\beta}{2} = \frac{2}{\sqrt[3]{3}}.
\]
Let \( \phi \) be the angle between the angle bisector and the perpendicular bisector of \( \overline{AB} \) drawn from angle \( \alpha \). Find \( \tan \phi \).

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