
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 $\sin\left(\alpha + \frac{\pi}{2}\right) + \cos(\pi - \alpha) = 0$,  
② The function $f(x) = \log_{3}(x^{2} - 2x)$ has a decreasing interval of $(-\infty, 1)$;  
③ Given $P: |2x-3| > 1$, $Q: \frac{1}{x^{2} + x - 6} > 0$, then $P$ is a necessary but not sufficient condition for $Q$;  
④ In the plane, the trajectory of the center of a moving circle that is tangent to both circles $x^{2} + y^{2} = 1$ and $x^{2} + y^{2} - 8x + 12 = 0$ is a hyperbola.  
The correct proposition numbers are __________.

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