
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 $f\left(x\right)=x^{3}-x^{2}-x+c$ (where $c$ is a constant),<br/>$(1)$ When $c=3$, find the equation of the tangent line to the function $f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$;<br/>$(2)$ Find the intervals where the function $f\left(x\right)$ is monotonic;<br/>$(3)$ Let $g\left(x\right)=[f\left(x\right)-x^{3}]\cdot e^{x}$. If the function $g\left(x\right)$ is monotonically increasing on the interval $\left[-3,2\right]$, find the range of real numbers for $c$.

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