
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 that the domain of the function $y=f(x)$ is $D$, if $∀x∈D,xf\left(x\right)≧f\left(x\right)$, then the function $f(x)$ is called an "$L$ type function".

(1) Determine whether the functions $y={e}^{x}$ and $y=\mathrm{ln}x$ are "$L$ type functions", and explain the reasons;

(2) Suppose the function $f(x)=(x+1)\mathrm{ln}x-(x-1)\mathrm{ln}a\phantom{\rule{0.22222em}{0ex}}(a>0)$, denoted as $g(x)={f}^{\mathrm{\prime }}(x)$.

① If the minimum value of the function $g(x)$ is $1$, find the value of $a$;

② If the function $f(x)$ is an "$L$ type function", find the range of values for $a$.

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