
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
Among the following propositions, the correct ones are \_\_\_\_\_\_.
① The function $y=\sin|x|$ is not a periodic function.  
② The function $y=\tan x$ is increasing within its domain.  
③ The period of the function $y=|\cos 2x+ \frac{1}{2}|$ is $\frac{\pi}{2}$.  
④ $y=\sin(x+ \frac{5\pi}{2})$ is an even function.  
⑤ The graph of the function $y=\sin(2x+ \frac{\pi}{3})$ is symmetric about the point ($\frac{\pi}{12}$, 0).

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