
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 $f(x)$ that satisfies $f(x+1)-f(x)=2x$ and $f(0)=1$.

1. Find the analytical expression of $f(x)$.
2. Determine the range of $y=f(x)$ on the interval $[-1,1]$.
3. Determine the range of $y=f(x)$ on the interval $[a,a+1]$.

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