
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
Given the following six propositions, the correct ones are __________  
① There exists $\alpha$ such that $\sin\alpha + \cos\alpha = \frac{3}{2}$;  
② $y = \sin\left(\frac{5}{2}\pi - 2x\right)$ is an even function;  
③ $x = \frac{\pi}{8}$ is a line of symmetry for $y = \sin(2x + \frac{5\pi}{4})$;  
④ $y = e^{\sin 2x}$ is an increasing function on $(0, \frac{\pi}{2})$ with a period of $\pi$;  
⑤ If $\alpha, \beta$ are angles in the first quadrant and $\alpha > \beta$, then $\tan\alpha > \tan\beta$;  
⑥ The graph of $y = 3\sin(2x + \frac{\pi}{3})$ can be obtained by shifting the graph of $y = 3\sin 2x$ to the left by $\frac{\pi}{3}$ units.

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