
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 vectors $m=(\sqrt{3}\cos x,-1)$, $n=(\sin x,\cos ^{2}x)$.

$(1)$ When $x=\frac{\pi}{3}$, find the value of $m\cdot n$;

$(2)$ If $x\in\left[ 0,\frac{\pi}{4} \right]$, and $m\cdot n=\frac{\sqrt{3}}{3}-\frac{1}{2}$, find the value of $\cos 2x$.

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