
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
Let the functions $f(x)=a\ln x-x$ and $g(x)=ae^{x}-x$, where $a$ is a positive real number.  
(Ⅰ) If $f(x)$ is a monotonically decreasing function on $(1,+\infty)$ and $g(x)$ has a minimum value on $(2,+\infty)$, find the range of $a$.  
(Ⅱ) If both functions $f(x)$ and $g(x)$ have no zeros, find the range of $a$.

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