
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 $f(x) = x + a \ln x$, if $\forall{x}_{1},{x}_{2} \in \left( \frac{1}{2}, 1 \right)\left({x}_{1}\neq {x}_{2}\right), \left|f\left({x}_{1}\right)-f\left({x}_{2}\right)\right| > \left| \frac{1}{{x}_{1}}- \frac{1}{{x}_{2}}\right|$, determine the range of the positive value $a$.

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