
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 $f(x)= \frac{2}{x}+a\ln x-2$ $(a > 0)$.

$(1)$ If the tangent line to the curve $y=f(x)$ at point $P(1,f(1))$ is perpendicular to the line $y=x+2$, find the intervals of monotonicity for the function $y=f(x)$;

$(2)$ If for all $x\in(0,+\infty)$, $f(x) > 2(a-1)$ holds, find the range of values for $a$;

$(3)$ Let $g(x)=f(x)+x-b$ $(b\in\mathbb{R})$. When $a=1$, the function $g(x)$ has two zeros in the interval $[e^{-1},e]$. Find the range of values for the real number $b$.

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