
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 $f(x)= \frac {2}{x}+a\ln x-2$, the tangent line to the curve $y=f(x)$ at point $P(1,f(1))$ is perpendicular to the line $y=x+3$.  
$(1)$ Find the value of the real number $a$;  
$(2)$ Let $g(x)=f(x)+x-b(b\in\mathbb{R})$, if the function $g(x)$ has two zeros in the interval $[e^{-1},e]$, find the range of the real number $b$;  
$(3)$ If the inequality $\pi^{f(x)} > \left( \frac {1}{\pi} \right)^{1+x-\ln x}$ holds for $|t|\leqslant 2$, find the range of the real number $x$.

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