
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 the function $f(x) = \tan(\omega x + \varphi) \left(0 < |\varphi| < \frac{\pi}{2}, \omega > 0\right)$ has a minimum positive period of $\frac{\pi}{2}$, and the graph of $f(x)$ passes through the point $\left(\frac{\pi}{3}, 0\right)$, determine the sum of all solutions to the equation $f(x) = \sin\left(2x + \frac{\pi}{3}\right) \left(x \in [0, \pi]\right)$.

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