
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
The quadratic function $f(x)$ satisfies $f(x+1) - f(x) = 2x$, and $f(0) = 1$.
   1. Find the analytic expression of $f(x)$.
   2. Determine the range of $f(x)$ over the interval $[-1, 1]$.
   3. Over the interval $[-1, 1]$, the graph of $y=f(x)$ is always above the graph of $y=2x+m$. Determine the range of the real number $m$.

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