
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) If $a < b < 0$, then $\dfrac{1}{a} < \dfrac{1}{b}$;  
(2) If $a > 0$, $b > 0$, then $\dfrac{a+b}{2} \geqslant \sqrt{ab} \geqslant \dfrac{ab}{a+b}$;  
(3) If $a < b < 0$, then $a^{2} > ab > b^{2}$;  
(4) $\lg 9 \cdot \lg 11 < 1$;  
(5) If $a > b$, $\dfrac{1}{a} > \dfrac{1}{b}$, then $a > 0$, $b < 0$;  
(6) For positive numbers $x$, $y$ satisfying $\dfrac{1}{x} + \dfrac{1}{y} = 1$, the minimum value of $x+2y$ is $6$.  
The correct proposition numbers are ______.

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