
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 $f(x) = x^2 - (2t + 1)x + t \ln x$ where $t \in \mathbb{R}$,

(1) If $t = 1$, find the extreme values of $f(x)$.

(2) Let $g(x) = (1 - t)x$, and suppose there exists an $x_0 \in [1, e]$ such that $f(x_0) \geq g(x_0)$ holds. Find the maximum value of the real number $t$.

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