
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 $f\left(x\right)=ae^{x}+\ln \left(ea\right)$, where $a \gt 0$, determine the range of real number $a$ if the inequality $f\left(x\right)\geqslant \ln \left(x-1\right)$ holds for any real number $x \gt 1$.

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