
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
Among the following propositions:
1. If $ac^2 > bc^2$, then $a > b$;
2. If $\sin \alpha = \sin \beta$, then $\alpha = \beta$;
3. "The real number $a = 0$" is the necessary and sufficient condition for "the line $x - 2ay = 1$ and the line $2x - 2ay = 1$ to be parallel";
4. If $f(x) = \log_2 x$, then $f(|x|)$ is an even function.

The correct propositions are numbered \_\_\_\_\_\_\_\_.

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