
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 interval in which the function $f(x) = \log_{\frac{1}{2}} (x^2 - 4)$ is decreasing is $(2, +\infty)$;
2. If a function $f(x)$ satisfies $f(x) = f(2-x)$, then the graph of $f(x)$ is symmetric about the line $x=1$;
3. The function $f'(x) = \log (x+1) + \log (x-1)$ is an even function;
4. If $f'(x)$ is the derivative of the function $f(x)$ and $f'(x_0) = 0$, then $x_0$ is an extreme point of $f(x)$.
The sequence numbers of all correct propositions are $\_\_\_\_\_\_\_\_$.

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