
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
For the function $f(x) = \cos 2x - 2 \sqrt {3} \sin x \cos x$, consider the following propositions:

1. If there exist $x_1$ and $x_2$ such that $x_1 - x_2 = \pi$, then $f(x_1) = f(x_2)$ holds;
2. $f(x)$ is monotonically increasing in the interval $\left[- \frac {\pi}{6}, \frac {\pi}{3}\right]$;
3. The graph of the function $f(x)$ is centrally symmetric about the point $\left( \frac {\pi}{12}, 0 \right)$;
4. If the graph of function $f(x)$ is shifted to the left by $\frac {5\pi}{12}$ units, it will coincide with the graph of $y = 2\sin 2x$.

Which of these propositions is/are correct? (Note: Fill in all the numbers that you think are correct)

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