
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
The function $f(x)=1+x- \frac {x^{2}}{2}+ \frac {x^{3}}{3}$, $g(x)=1-x+ \frac {x^{2}}{2}- \frac {x^{3}}{3}$, if the function $F(x)=f(x+3)g(x-4)$, and the zeros of the function $F(x)$ are all within $[a,b]$ $(a < b$, $a$, $b\in\mathbb{Z})$, then the minimum value of $b-a$ is \_\_\_\_\_\_.

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