
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 $f(x) = ax^{2} + bx + c \, (a \neq 0)$ satisfies $f(x+1) - f(x) = 2x$, and $f(0) = 1$.
$(1)$ Find the expression of $f(x)$;
$(2)$ If the inequality $f(x) > 2x + m$ always holds in the interval $[-1,1]$, find the range of the real number $m$.

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