
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
Find all differentiable functions \( f: \mathbf{R} \longrightarrow \mathbf{R} \) satisfying the identity

\[ 
f^{\prime}\left(\frac{x+y}{2}\right)=\frac{f(y)-f(x)}{y-x}, \quad x, y \in \mathbf{R}, \quad x \neq y 
\]

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