
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 $f(x) = x \ln x$.
(1) Find the intervals of monotonicity and the minimum value of the function $y = f(x)$;
(2) If the function $F(x) = \frac{f(x) - a}{x}$ has a minimum value of $\frac{3}{2}$ on the interval $[1, e]$, find the value of $a$;
(3) If $k \in \mathbb{Z}$, and $f(x) + x - k(x - 1) > 0$ for any $x > 1$, find the maximum value of $k$.

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