
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 $f(x)=\ln x-a+\frac{a}{x}$ where $a>0$.<br/>$(Ⅰ)$ If the curve $y=f\left(x\right)$ is tangent to the $x$-axis at the point $\left(1,f\left(1\right)\right)$, find the value of $a$;<br/>$(Ⅱ)$ Find the number of zeros of the function $f\left(x\right)$ on the interval $\left(1,e\right)$;<br/>$(Ⅲ)$ If for all $x_{1}$, $x_{2}\in \left(1,e\right)$, $(x_{1}-x_{2})(|f(x_{1})|-|f(x_{2})|) \gt 0$, write down the range of values for $a$. (Only state the conclusion)

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