
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 $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$, where $\overrightarrow{a} = (\sqrt{3}, 2\sin \frac{\omega x}{2})$ and $\overrightarrow{b} = (\sin \omega x, -\sin \frac{\omega x}{2})$, $\omega > 0$, and the smallest positive period of $f(x)$ is $\pi$.

(1) Find the value of $\omega$;  
(2) Find the minimum value of $f(x)$ and the corresponding set of values of $x$;  
(3) If the graph of $f(x)$ is translated to the left by $\varphi$ units, and the resulting graph is symmetric about the point $(\frac{\pi}{3}, 0)$, find the smallest positive value of $\varphi$.

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