
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 the equation $f(x+1) - f(x-1) = 4x + 2$ for any real number $x$, and $f(1) = 3$.

1. Find the analytic expression of the function $f(x)$.
2. The function $g(x) = f(x) - (1 + 2m)x + 1$ ($m \in \mathbb{R}$) has a minimum value of $-3$ on the interval $[2, +\infty)$. Find the value of the real number $m$.

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