
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 $f(x)=ax-\ln x$, where $a\in\mathbb{R}$.  
(I) When $a=2$, find the equation of the tangent line to the curve $f(x)$ at the point $(1,f(1))$.  
(II) If $f(x)$ has an extremum at $x=1$, find the interval of monotonic increase for $f(x)$.  
(III) If the minimum value of $f(x)$ in the interval $(0,e]$ is $3$, find the value of $a$.

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