
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
There are the following four propositions:  
① The function $f(x) = \sin\left(\frac{\pi}{3} - 2x\right)$ has an increasing interval of $\left[\frac{\pi}{4}, \frac{\pi}{3}\right]$;  
② If the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$) is an odd function, then $\varphi$ is an integer multiple of $\pi$;  
③ For the function $f(x) = \tan\left(2x + \frac{\pi}{3}\right)$, if $f(x_1) = f(x_2)$, then $x_1 - x_2$ must be an integer multiple of $\pi$;  
④ The graph of the function $y = 2\sin\left(2x + \frac{\pi}{3}\right)$ is symmetric about the point $\left(\frac{\pi}{3} + \frac{k\pi}{2}, 0\right)$.  
Among these propositions, the correct ones are \_\_\_\_\_\_ (fill in the correct proposition numbers).

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