
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 $f\left( x \right)=\frac{mx}{{{x}^{2}}+n}\left( m,n\in R \right)$ that reaches an extreme value of $2$ at $x=1$.

(1) Find the analytical expression of $f\left( x \right)$;

(2) Let $g\left( x \right)=\ln x+\frac{a}{x}$. If for any $x_1\in \left[ -1,1 \right]$, there exists $x_2\in \left[ 1,e \right]$ such that $g\left( {{x}_{2}} \right)\leqslant f\left( {{x}_{1}} \right)+\frac{7}{2}$ holds true, find the range of values of the real number $a$.

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