
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 real functions \( f, g, h: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), the following inequalities hold:
$$
(x - y)f(x) + h(x) - xy + y^2 \leq h(y) \leq (x - y)g(x) + h(x) - xy + y^2.
$$

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