
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 functions $f\left(x\right)=\frac{{2alnx}}{x}+x$ and $g\left(x\right)=2x-\frac{1}{x}$, where $a\in R$.  
$(1)$ If the equation $f\left(x\right)=g\left(x\right)$ has a unique real number solution in the interval $\left[1,e\right]$ ($e$ is the base of the natural logarithm), find the range of real numbers for $a$.  
$(2)$ If there exists a point $t_{0}$ in the interval $\left[1,e\right]$ such that the inequality $xf\left(x\right) \gt x^{2}+\frac{{2({a+1})}}{x}+2x$ holds for $x$, find the range of real numbers for $a$.

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