
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 the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies:
\[ x[f(x+1)-f(x)] = f(x), \forall x \in \mathbf{R} \]
and
\[ |f(x)-f(y)| \leq |x-y|, \forall x, y \in \mathbf{R}. \]

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