
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 $f(x)=\cos 2x-2 \sqrt {3}\sin x\cos x$, determine the correct proposition number(s) from the following options:
  
$①$ For any $x\_1$ and $x\_2$, if $x\_1-x\_2=π$, then $f(x\_1)=f(x\_2)$ holds true;
  
$②$ The function $f(x)$ is monotonically increasing in the interval $[- \frac {π}{6}, \frac {π}{3}]$;
  
$③$ The graph of the function $f(x)$ is symmetrical about the point $( \frac {π}{12},0)$;
  
$④$ After shifting the graph of the function $f(x)$ to the left by $ \frac {5π}{12}$ units, it will coincide with the graph of $y=\sin 2x$.

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