
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
Given a function $y=f(x)$ defined on domain $D$, if it satisfies the following conditions simultaneously: $(1)$ $f(x)$ is monotonically increasing or decreasing in $D$; $(2)$ 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 $y=f(x)$, $x \in D$ a closed function.
$(1)$ Find the interval $[a,b]$ where the closed function $y=-x^{3}$ satisfies condition $(2)$;
$(2)$ If function $y=k+\sqrt{x+2}$ is a closed function, find the range of values for the real number $k$.

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