
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 \( f(x) \) be a convex function defined on an open interval \( (a, b) \), and let \( x_{1}, x_{2}, x_{3}, x_{4} \in (a, b) \) with \( x_{1} < x_{2} < x_{3} < x_{4} \). Then,
$$
\begin{array}{c}
\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}} \leqslant f^{\prime}-\left(x_{2}\right) \leqslant f^{\prime}+\left(x_{2}\right) \leqslant \\
f^{\prime}-\left(x_{3}\right) \leqslant f^{\prime}+\left(x_{3}\right) \leqslant \frac{f\left(x_{4}\right)-f\left(x_{3}\right)}{x_{4}-x_{3}}
\end{array}
$$

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