
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 $f(x)$ satisfies $f(x+1)+f(x-1)=2x^{2}-2x$.

$(1)$ Solve the inequality $f(x) > 1$;                  

$(2)$ If $x \in [0,2]$, find the range of the function $f(x)$.

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