
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 that $f(x)$ is a quadratic function, and $f(0)=1$, $f(x+1)-f(x)=2x$.  
$(1)$ Find the expression for $f(x)$;  
$(2)$ If for any $x\in [1,+\infty)$, $g(x)=f(x)-a > 0$ always holds, find the range of real number $a$.

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