
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 functions $f\left(x\right)=\ln x$ and $g\left(x\right)=e^{x}$, $\exists x_{1}$, $x_{2}\in \left[1,2\right]$ such that $|g(x_{1})-g(x_{2})| \gt k|f(x_{1})-f(x_{2})|$, determine the range of constant $k$.

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