
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
Let \( f: \mathbf{R}_{+} \rightarrow \mathbf{R} \) satisfy, for any \( x, y \in \mathbf{R}_{+} \):
\[ \frac{f(\sqrt{x})+f(\sqrt{y})}{2}=f\left(\sqrt{\frac{x+y}{2}}\right) \]

Then, for any \( x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}_{+} \), we have:
\[ \frac{1}{n} \sum_{k=1}^{n} f\left(\sqrt{x_{k}}\right)=f\left(\sqrt{\frac{1}{n} \sum_{k=1}^{n} x_{k}}\right) \]

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