
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 $f(x)=\tan\left( \frac{1}{3}x - \frac{\pi}{6}\right)$:
1. Find the smallest positive period of $f(x)$.
2. Find the value of $f\left( \frac{3\pi}{2} \right)$.
3. If $f\left(3\alpha+ \frac{7\pi}{2}\right)=-\frac{1}{2}$, find the value of $\frac{\sin(\pi-\alpha)+\cos(\alpha-\pi)}{\sqrt{2}\sin\left(\alpha+ \frac{\pi}{4}\right)}$.

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