
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 the function $f(x)=\left\{{\begin{array}{l}{{e^x}-x,x≤k,}\\{{x^3}-x+1,x＞k．}\end{array}}\right.$<br/>$(1)$ If $k=0$, the solution set of the inequality $f\left(x\right) \lt 1$ is ______;<br/>$(2)$ If the function $g\left(x\right)=f\left(x\right)-1$ has exactly two zeros, then the range of real number $k$ is ______.

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