
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 $f(x) = ax + x\ln x$ where $a \in \mathbb{R}$:
1. When $a = 2$, determine the intervals on which the function $f(x)$ is increasing and decreasing.
2. When $a = 1$ and $k \in \mathbb{Z}$, find the maximum value of $k$ such that the inequality $k(x-1) < f(x)$ holds for all $x \in (1, +\infty)$.

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