
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
Suppose the function \( f(x) \) for the interval \( (a, b) \) [note: this interval can be open, closed, or half-open and half-closed] has the property for any two sets of values \( x_{1}, x_{2} \) and \( y_{1}, y_{2} \) in the interval with \( x_{1} + x_{2} = y_{1} + y_{2} \) that:
\[
\begin{array}{l}
f\left(x_{1}\right) + f\left(x_{2}\right) < f\left(y_{1}\right) + f\left(y_{2}\right) \\
\Leftrightarrow \left| y_{1} - y_{2} \right| < \left| x_{1} - x_{2} \right|,
\end{array}
\]

then for any set of numbers \( x_{1}, x_{2}, \cdots, x_{n} \) in \( (a, b) \), we have
\[
f\left(x_{1}\right) + f\left(x_{2}\right) + \cdots + f\left(x_{n}\right) \leqslant n f\left(\frac{x_{1} + x_{2} + \cdots + x_{n}}{n}\right).
\]

Furthermore, the equality in the above expression holds if and only if \( x_{1} = x_{2} = \cdots = x_{n} \).

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