
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
The propositions given are:  
(1) The function $y=\cos \left( \frac{2}{3}x+ \frac{7\pi}{2}\right)$ is an odd function;  
(2) If $\alpha$ and $\beta$ are both first quadrant angles and $\alpha < \beta$, then $\tan \alpha < \tan \beta$;  
(3) The function $y=2\sin \left( \frac{2}{3}x+ \frac{\pi}{3}\right)$ has a minimum value of $-2$ and a maximum value of $\sqrt{3}$ on the interval $[-\pi, \frac{\pi}{2}]$;  
(4) The line $x= \frac{\pi}{8}$ is a symmetry axis of the graph of the function $y= \frac{1}{2}\sin (5x+ \frac{7\pi}{8})$.  
The correct proposition numbers are ______. (Write down all the correct proposition numbers)

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