
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 functions $f(x)=1+x- \frac{x^{2}}{2}+ \frac{x^{3}}{3}$ and $g(x)=1-x+ \frac{x^{2}}{2}- \frac{x^{3}}{3}$, define the function $F(x)=f(x) \cdot g(x)$. If all the zero points of function $F(x)$ are within the interval $[a,b]$ where $a,b \in \mathbb{Z}$ and $a < b$, find the minimum value of $b-a$.

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