
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 $f(x)=\ln{x}-\frac{1}{x}-2x$.
(1) Find the monotonic intervals of $f(x)$;
(2) If the line $l: y=ax+b$ is a tangent line of the function $F(x)=f(x)+2x$ and $a,b\in\mathbb{R}$, find the minimum value of $a+b$.

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