
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 simultaneously satisfies: (1) $f(x+1) - f(x) = 2x$; (2) For all $x \in \mathbb{R}$, the inequality $f(x) \geq x^2 - x + 1$ holds; (3) When $x \geq 0$, $f(x) \leq 2^x$.  
(a) Find the explicit formula for $f(x)$;  
(b) For $x \in [-1, 1]$, the inequality $f(x) > 2x + m$ always holds, find the range of the real number $m$.

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