
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 $f(x) = x^3 - 3ax$ ($a \in \mathbb{R}$),  
(1) When $a=1$, find the minimum value of $f(x)$;  
(2) If the line $x+y+m=0$ is not a tangent to the curve $y=f(x)$ for any $m \in \mathbb{R}$, find the range of $a$;  
(3) Let $g(x) = |f(x)|$, $x \in [-1, 1]$, find the expression for the maximum value $F(a)$ of $g(x)$.

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