
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 $b>0$, let $x_{1}= \frac{a+b}{2}, x_{2}= \sqrt{ab}, x_{3}= \sqrt{\frac{a^{2}+b^{2}}{2}}$. The relationship in terms of size between $x_1, x_2, x_3$ is: ___________ (Please link the three numbers $x_1, x_2, x_3$ with the inequality symbol “≥”).

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