
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 The smallest positive period of $y=f(x)$ is $\pi$.  
③ $y=f(x)$ is a decreasing function in the interval $(\frac{\pi}{24}, \frac{13\pi}{24})$.  
④ After shifting the graph of the function $y= \sqrt{2}\cos2x$ to the right by $\frac{\pi}{24}$ units, it will coincide with the graph of the given function.  
Among these, the correct propositions are \_\_\_\_\_\_.

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