
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 $f\left( x \right)=\frac{\ln x}{x}-\frac{k}{x} \left( k\in R \right).$

(1) If the slope of the tangent line to the curve $y=f\left( x \right)$ at the point $\left( 1,f\left( 1 \right) \right)$ is $10$, find the maximum value of the function $f\left( x \right)$;

(2) If the inequality ${{x}^{2}}f\left( x \right)+\frac{1}{x+1}\geqslant 0$ and $k\geqslant \frac{1}{2}{{x}^{2}}+({{e}^{2}}-2)x-{{e}^{x}}-7$ hold true on $\left[ 1,+\infty \right)$, find the range of the real number $k$.

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