
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 the interval $\left[a,b\right]$ satisfies: ① The function $f\left(x\right)$ is defined and monotonic on $\left[a,b\right]$; ② The range of the function $f\left(x\right)$ on $\left[a,b\right]$ is also $\left[a,b\right]$, then the interval $\left[a,b\right]$ is called the resonance interval of the function $f\left(x\right)$. Please complete:<br/>$(1)$ Write down a resonance interval for the function $f\left(x\right)=x^{3}$______;<br/>$(2)$ If the function $f\left(x\right)=\sqrt{x+1}-k$ has a resonance interval, then the range of the real number $k$ is______.

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