
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 $f(x)$ which satisfies $f(x+1) - f(x) = 2x$ for all real numbers $x$, and $f(0) = 1$.  
(1) Find the expression for $f(x)$.  
(2) Determine the range of $f(x)$ when $x \in [-2, 4]$.

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