
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 four propositions:
1. If $a > b > 0$, then $\frac{1}{a} < \frac{1}{b}$.
2. If $a > b > 0$, then $a^2 - a > b^2 - b$.
3. If $a > b > 0$, then $a^3 > b^3$.
4. If $a > 0$, $b > 0$, and $2a + b = 1$, then the minimum value of $a^2 + b^2$ is 9.

Among these propositions, the correct ones are \_\_\_\_\_\_．(Fill in the numbers of all propositions you think are correct)

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