
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
If the quadratic function $f(x) = ax^2 + bx + c$ ($a, b \in \mathbb{R}$) satisfies $f(x + 1) - f(x) = 2x$ and $f(0) = 1$,

1. Find the expression for $f(x)$.
2. If the inequality $f(x) \geq 2x + m$ always holds in the interval $[-1, 1]$, determine the range of the real number $m$.

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