
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 $f(x)=2\sin \left(\omega x- \frac{\pi}{6}\right)\cdot \cos \omega x+ \frac{1}{2}$ where $\omega > 0$, its smallest positive period is $\pi$.

(1) Find the value of $\omega$;

(2) Move the graph of the function $y=f(x)$ to the left by $\frac{\pi}{6}$ units, then stretch the horizontal coordinates of the obtained graph by a factor of 2 while keeping the vertical coordinates unchanged, to get the graph of the function $g(x)$. Find the zeros of the function $g(x)$ on the interval $[-\pi,\pi]$.

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