
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 the function $f(x)=ax^2+bx+c (a\neq 0)$, its graph passes through the point $(0,1)$, and it has a unique zero point $(-1)$.

(1) Find the expression of $f(x)$;

(2) When $x\in [-1,1]$, find the minimum value $g(k)$ of the function $F(x)=f(x)-kx$.

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