
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
Regarding the following propositions:

1. The function $y=\tan x$ is increasing in the first quadrant;
2. The function $y=\cos 2(\frac{\pi}{4}-x)$ is an even function;
3. The function $y=4\sin (2x-\frac{\pi}{3})$ has a symmetry center at $(\frac{\pi}{6},0)$;
4. The function $y=\sin (x+\frac{\pi}{4})$ is increasing on the closed interval $[-\frac{\pi}{2},\frac{\pi}{2}]$;

Write down the numbers of all correct propositions: $\_\_\_\_\_\_\_\_$.

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