
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 $f(x)=x^{2}-2x-a\ln x$, $g(x)=ax$.  
$(1)$ Find the extreme values of the function $F(x)=f(x)+g(x)$.  
$(2)$ If the inequality $\dfrac {\sin x}{2+\cos x}\leqslant g(x)$ always holds for $x\geqslant 0$, find the range of values for $a$.

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