
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 $f(x)$ that satisfies $f(x+1) - f(x) = 2x$ and $f(0) = 1$:
1. Find the expression for $f(x)$;
2. If $x \in [-1, 1]$, determine the range of real numbers $m$ such that the inequality $f(x) > 2x + m$ always holds;
3. Let $g(a)$ be the minimum value of the function $f(x)$ in the interval $[a, a+1]$, find the expression for $g(a)$.

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