
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
Given that $α \in (0, \frac{π}{2})$ and $β \in (0, \frac{π}{2})$, if $\tan (α + β) = 2 \tan β$, then when $α$ takes the maximum value, $\tan 2α = \_\_\_\_\_\_$.

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