
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. For all $x \in \mathbb{R}$, where $x \neq 0$, it holds that $x + \frac{1}{x} \geq 2$.
2. There exists an $x \in \mathbb{R}$ such that $x^2 + 1 \leq 2x$.
3. If $x > 0$ and $y > 0$, then $\sqrt{\frac{x^2 + y^2}{2}} \geq \frac{2xy}{x + y}$.

Identify all true propositions by their numbers.

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