
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 extreme points of the function $f(x)$.
2. Let $g(x) = f(x) - a(x-1)$, where $a \in \mathbb{R}$. Find the minimum value of the function $g(x)$ on the interval $[1,e]$ ($e=2.71828\ldots$).

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