
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 propositions:

1. The function $y=\sin \left( \frac{5\pi}{2}-2x\right)$ is an even function;
2. The function $y=\sin \left(x+ \frac{\pi}{4}\right)$ is increasing on the closed interval $\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$;
3. The line $x= \frac{\pi}{8}$ is an axis of symmetry for the graph of the function $y=\sin \left(2x+ \frac{5\pi}{4}\right)$;
4. Translating the graph of the function $y=\cos \left(2x- \frac{\pi}{3}\right)$ to the left by $\frac{\pi}{3}$ units results in the graph of the function $y=\cos 2x$;

The correct propositions are: \_\_\_

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