
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 $f(x)=ax^{2}+bx+c$ where $a,b \in \mathbb{R}$, it satisfies $f(x+1)-f(x)=2x$, and $f(0)=1$.  
$(1)$ Find the expression for $f(x)$;  
$(2)$ If the inequality $f(x) \geqslant 2x+m$ 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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