
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
If for any two distinct positive real numbers $x_{1}$ and $x_{2}$, $\frac{f\left(x_{1}\right)-f\left(x_{2}\right)}{x_{1}-x_{2}}\gt 2$, where $f\left(x\right)=\ln x+ax^{2}$, determine the range of real number $a$.

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