
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 $g(x) = 1 - x + \frac{x^2}{2} - \frac{x^3}{3}$, let the function $F(x) = f(x) \cdot g(x)$. If the zeros of the function $F(x)$ are all within the interval $[a, b]$ ($a < b$, $a, b \in \mathbb{Z}$), then the minimum value of $b-a$ is \_\_\_\_\_.

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