
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 $f(-1+k)=f(-1-k)$, where $k\in R$, and the graph of the function intersects the $y$-axis at point $(0,1)$, and the length of the line segment intercepted on the $x$-axis is $2\sqrt{2}$. Find the analytical expression of the quadratic function.

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