
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 $f(x)=a(x- \frac {1}{x})-2\ln x(a\in\mathbb{R})$.  
$(1)$ If $a=2$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;  
$(2)$ Determine the intervals of monotonicity for the function $f(x)$;  
$(3)$ Let the function $g(x)=- \frac {a}{x}$. If there exists at least one $x_{0}\in[1,e]$ such that $f(x_{0}) > g(x_{0})$ holds, find the range of the real number $a$.

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