
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
If a function \( f(x) \) with domain \( D \) being the set of real numbers \( \mathbf{R} \) satisfies the following two conditions:
(1) \( f(x) \) is either a monotonically increasing function or a monotonically decreasing function on \( D \);
(2) There exists an interval \([a, b] \subseteq D\) such that the range of \( f(x) \) on \([a, b]\) is also \([a, b]\),
then we call such a function a "closed function."

(1) Is the function \( y = -x^3 \) with domain \(\mathbf{R}\) a "closed function"? Please explain your reasoning.

(2) Given the function \( f(x) = \frac{x^2}{2} - x + 1 \) for \( x \in [a, b] \), can it be a "closed function"? Please explain your reasoning.

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