
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 $a$ and $b$ are constants, with $a \neq 0$, the function $f(x) = -ax + b + ax \ln x$, and $f(e) = 2$ (where $e = 2.71828...$ is the base of the natural logarithm).

(1) Find the value of the real number $b$;

(2) Determine the monotonic intervals of the function $f(x)$.

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