
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 $f(x)=x^{2}-(2t+1)x+t\ln x\ (t\in\mathbb{R})$,
(1) If $t=1$, find the extreme values of $f(x)$;
(2) Let $g(x)=(1-t)x$. If there exists $x_{0}\in[1,e]$ such that $f(x_{0})\geqslant g(x_{0})$ holds, find the maximum value of the real number $t$.

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