
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 $f(0) = 1$ and $f(x + 1) - f(x) = 2x + 2$.
(I) Find the analytic expression for $f(x)$;
(II) If $g(x) = 2^{f(x)}$, where $x \in [-1, 1]$, find the range of $g(x)$.

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