
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 $f(x)=2ax-\frac{b}{x}+\ln x$, if $f(x)$ takes extreme values at $x=1$, $x=\frac{1}{2}$,
(i) find the values of $a$ and $b$;
(ii) there exists $x\_0$ in $[\frac{1}{4},2]$ such that the inequality $f(x\_0)-c\leqslant 0$, find the minimum value of $c$.

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