
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 $f(x)=a\ln x-2x$ and $g(x)=f(x)+x^{2}$.  
$(1)$ If $a=2e$, find the maximum value of the function $f(x)$.  
$(2)$ When $a \gt 0$, if the function $g(x)$ has two critical points $x_{1}$ and $x_{2}$ ($x_{1} \lt x_{2}$), and the inequality $g(x_{1})\geqslant \lambda x_{2}$ always holds, find the range of the real number $\lambda$.

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