
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 that the domain of the function $f(x)$ is $R$, and the graph always passes through the point $(0,2)$. For any $x_{1}$, $x_{2} \in R$, when $x_{1} \neq x_{2}$, it is always true that $\frac{{f(x_{1}) - f(x_{2})}}{{x_{1} - x_{2}}} > 1$. Find the solution set of the inequality $f[\ln (e^{x}-2)] < 2+\ln (e^{x}-2)$.

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