
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
If for any $x, y \in \mathbf{R}$, the inequality $f(x) + f(y) \leqslant 2f\left(\sqrt[4]{\frac{x^{4} + y^{4}}{2}}\right)$ holds, then for any integer $n \geqslant 2$ and $x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}$, the following inequality holds:
$$f(x_1) + f(x_2) + \cdots + f(x_n) \leqslant n f\left[\sqrt{\frac{x_{1}^{k} + x_{2}^{k} + \cdots + x_{n}^{k}}{n}}\right].$$

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