
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 $f\left(x\right)=2\tan \left(\omega x+\varphi \right)$ where $(\omega > 0$, $|\varphi | < \frac{π}{2})$, $f\left(0\right)=\frac{{2\sqrt{3}}}{3}$, and the period $T\in (\frac{π}{4}$, $\frac{3π}{4})$, the point $(\frac{π}{6}$, $0)$ is the center of symmetry of $f\left(x\right)$. Evaluate the function $f\left(x\right)$ at $x=\frac{π}{3}$.

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