
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
Let $f(x)=\sqrt{3}\cos(2\pi-x)-\cos\left(\frac{\pi}{2}+x\right)+1$.  
$(1)$ Find the range and the interval of monotonic increase of the function $f(x)$.  
$(2)$ When $f(\alpha)=\frac{13}{5}$ and $\frac{\pi}{6}<\alpha<\frac{2\pi}{3}$, find the value of $\cos 2\alpha$.

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