
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 $f(x) = ax + \ln x$, where $x \in [1, +\infty)$,  
(1) If $f'(x_0) = \frac{f(e) - f(1)}{e - 1}$, find the value of $x_0$;  
(2) If the function $f(x)$ is monotonically decreasing on $[1, +\infty)$, find the range of values for $a$.

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