
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 $f\left( x \right)=a{{x}^{2}}+bx+c\left( a\ne 0 \right)$ that satisfies $f\left( x+1 \right)-f\left( x \right)=2x$, and $f\left( 0 \right)=1$.

(1) Find the analytic expression of $f\left( x \right)$;

(2) If the inequality $f\left( x \right) > 2x+m$ always holds true in the interval $\left[-1,1 \right]$, find the range of values for the real number $m$.

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