
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 $f(x) = \ln x - ax + b$, where $a, b \in \mathbb{R}$.

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

(Ⅱ) If $a=1$, $b \in [0, 2]$, and there exists a real number $k$, such that for any real number $x \in [1, e]$, it always holds that $f(x) \geq kx - x\ln x - 1$, find the maximum value of $k-b$.

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