
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 function $f(x) = ax^3 + bx^2 + cx$ where $a \neq 0$, and $6a + b = 0$, $f(1) = 4a$.  
$(1)$ Discuss the monotonicity of $f(x)$.  
$(2)$ If $x \in [0,3]$, the function $F(x) = f(x) - xe^{-x}$ has three zeros $x_1$, $x_2$, $x_3$ with $x_1 < x_2 < x_3$. Compare $x_1 + x_2 + x_3$ with $2$ and explain the reason.

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