
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 $\overrightarrow{a}=(\sin x, \sqrt{3}\cos x)$ and $\overrightarrow{b}=(\cos x,-\cos x)$, the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}+ \frac{ \sqrt{3}}{2}$.

$(1)$ Find the equation of the axis of symmetry for the graph of $y=f(x)$;

$(2)$ If the solution of the equation $f(x)= \frac{1}{3}$ in the interval $(0,\pi)$ is ${x}_1$ 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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