
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 $f(x)=\sin \omega x\cos \omega x- \sqrt{3}\cos^2\omega x+ \frac{\sqrt{3}}{2} (\omega > 0)$, the two adjacent axes of symmetry of its graph are $\frac{\pi}{2}$.

$(1)$ Find the equation of the axis of symmetry for the function $y=f(x)$.

$(2)$ If the zeros of the function $y=f(x)- \frac{1}{3}$ in the interval $(0,\pi)$ are $x_{1}$ and $x_{2}$, find the value of $\cos (x_{1}-x_{2})$.

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