
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
Let the function $f(x)=\ln(x+(x-a)^2)$, where $a\in\mathbb{R}$.

(I) If $a=0$, find the minimum value of the function $f(x)$ on the interval $[1,e]$;

(II) If the function $f(x)$ has an increasing interval on $[\frac{1}{2},2]$, find the range of the real number $a$;

(III) Find the extreme points of the function $f(x)$.

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