
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 $f(x)= \frac {a\ln x+b}{x}$ with $a\leqslant 2$ and $a\neq 0$, the tangent line to the function $f(x)$ at point $(1,f(1))$ passes through the point $(3,0)$.  
(1) Find the relation between $a$ and $b$ and discuss the monotonicity intervals of the function $f(x)$;  
(2) Given $g(x)=x+ \frac {2}{x}-a-2$, suppose the function $F(x)=f(x)+g(x)$ has exactly one zero in the interval $(0,2\]$. Find the range of the real number $a$.

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