
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 $f(x)=ax(\ln x-1)-x^{2}$ where $a\in \mathbb{R}$, it has exactly two extremum points, $x_{1}$ and $x_{2}$, and $x_{1}<x_{2}$.

(Ⅰ) Find the range of real numbers $a$;

(Ⅱ) If the inequality $\ln x_{1}+\lambda\ln x_{2} > 1+\lambda$ always holds, find the range of real number $\lambda$.

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