
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 $f(x)=\tan(\omega x+\phi)$ $(\omega>0, 0<|\phi|<\frac{\pi}{2})$, where two adjacent branches of the graph intersect the coordinate axes at points $A(\frac{\pi}{6},0)$ and $B(\frac{2\pi}{3},0)$. Find the sum of all solutions of the equation $f(x)=\sin(2x-\frac{\pi}{3})$ for $x\in [0,\pi]$.

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