
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 quadratic function $f\left(x\right)$ that satisfies $f\left(x+1\right)-f\left(x\right)=2x$, and $f\left(0\right)=-1$.<br/>$(1)$ Find the analytical expression of $f\left(x\right)$;<br/>$(2)$ Find the range of $g\left(x\right)=xf\left(x\right)$ for $x\in \left[-1,2\right]$.

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