
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]


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Regarding the function $f\left( x \right)=\cos 2x-2\sqrt{3}\sin x\cos x$, among the following propositions, the correct proposition number is ______.

$①$ If there exist $x_{1}$, $x_{2}$ such that $x_{1}-x_{2}=\pi$, then $f\left( x_{1} \right)=f\left( x_{2} \right)$ holds  
$②$ $f\left( x \right)$ is monotonically increasing in the interval $\left[ -\frac{\pi }{6},\frac{\pi }{3} \right]$  
$③$ The graph of the function $f\left( x \right)$ is centrally symmetric about the point $\left( \frac{\pi }{12},0 \right)$  
$④$ After shifting the graph of the function $f\left( x \right)$ to the left by $\frac{5\pi }{12}$ units, it will coincide with the graph of $y=2\sin 2x$

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