
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
When $a > 0$ and $b > 0$, the following inequalities always hold: $\boxed{1} (a+b)(\frac{1}{a} + \frac{1}{b}) \geq 4$, $\boxed{2} a^2 + b^2 + 2 \geq 2a + 2b$, $\boxed{3} \sqrt{|a-b|} \geq \sqrt{a} - \sqrt{b}$, and $\boxed{4} \frac{2ab}{a+b} \geq \sqrt{ab}$.

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