
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 $f(x)$ satisfying $f(x+1) - f(x) = 2x$ for all $x \in \mathbb{R}$, and $f(0)=1$,  
(1) Find the expression of $f(x)$;  
(2) Find the range of the function $g(x) = f(x) - 2x$ when $x \in [-1,1]$.

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