
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 that $a > 0$ and $b > 0$, determine the correct conclusion(s) from the following options:

$① \frac{2ab}{a+b} \leqslant \frac{a+b}{2}$;

$② \sqrt{ab} \leqslant \frac{a+b}{2}$;

$③ \frac{a+b}{2} \leqslant \sqrt{\frac{a^{2}+b^{2}}{2}}$;

$④ \frac{b^{2}}{a} + \frac{a^{2}}{b} \geqslant a + b$.

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