
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 $f(x)=x^3-3ax (a\in R)$ and $g(x)=\ln x$,

(1) When $a=1$, find the equation of the tangent line of $t(x)=g(x)-f(x)$ at $x=1$;

(2) If the graph of $f(x)$ is always above the graph of $g(x)$ in the interval $[1,2]$, find the range of $a$;

(3) Let $h(x)=|f(x)|, x\in [-1,1]$, find the maximum value $F(a)$ of $h(x)$ in terms of $a$.

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