
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
The domain of the function $f(x) = (x-1)^2 + a\ln x$ is $({\frac{1}{4}, +∞})$, and there are exactly two extreme points $x_{1}$, $x_{2}(x_{1} \lt x_{2})$ within the domain.<br/>$(1)$ Find the range of real number $a$;<br/>$(2)$ If $f(x_{2}) \gt \lambda x_{1}$ always holds, find the range of real number $\lambda$.

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