
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 __________. (Write down the numbers of all correct propositions.)

① There exists an $\alpha$ satisfying $\sin \alpha + \cos \alpha = \frac{3}{2}$;

② $y = \cos \left( \frac{7\pi}{2} -3x \right)$ is an odd function;

③ A center of symmetry for $y = 4\sin \left(2x + \frac{5\pi}{4} \right)$ is $\left(- \frac{9\pi}{8} ,0\right)$;

④ The graph of $y = \sin \left(2x - \frac{\pi}{4} \right)$ can be obtained by shifting the graph of $y = \sin 2x$ to the right by $\frac{\pi}{4}$ units.

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