
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
Regarding the function $f(x)=\cos (2x- \frac {\pi}{3})+\cos (2x+ \frac {\pi}{6})$, the following propositions are given:  
$(1)$ The maximum value of $y=f(x)$ is $\sqrt {2}$;  
$(2)$ $y=f(x)$ is a periodic function with the smallest positive period of $\pi$;  
$(3)$ $y=f(x)$ is monotonically decreasing in the interval $( \frac {\pi}{24}, \frac {13\pi}{24})$;  
$(4)$ After shifting the graph of the function $y= \sqrt {2}\cos 2x$ to the left by $\frac {\pi}{24}$ units, it will coincide with the graph of the given function.  
Among these, the correct proposition numbers are ______. (Note: Fill in all the proposition numbers you think are correct.)

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