
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 $\dfrac{1}{a} > \dfrac{1}{b}$;  
(2) If $a > b > 0$, then $a - \dfrac{1}{a} > b - \dfrac{1}{b}$;  
(3) If $a > b > 0$, then $\dfrac{2a+b}{a+2b} > \dfrac{a}{b}$;  
(4) If $a > 0$, $b > 0$ and $2a+b=1$, then the minimum value of $\dfrac{2}{a} + \dfrac{1}{b}$ is $9$.  
Among these, the correct propositions are ______ (Fill in the numbers of all correct propositions).

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