
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 the function $f(x)=\tan (\omega x + \frac{\pi}{4})$ with $\omega > 0$, the smallest positive period is $\frac{\pi}{2}$.  
(I) Find the value of $\omega$ and the domain of the function $f(x)$.  
(II) If $f\left( \frac{\alpha}{2} \right)=3$, find the value of $\tan 2\alpha$.

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