
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 $f(x) = ax^2 + bx + c$, it satisfies the conditions $f(0) = 0$ and $f(x+2) - f(x) = 4x$.
(I) Find the analytic expression of function $f(x)$;
(II) If function $g(x) = f(x) - 2mx + 2$, find the minimum value of function $g(x)$ when $x \in [1, +\infty)$.

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