
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 $f(x)=2ax- \frac{b}{x}+\ln x$ has extreme values at $x=1$ and $x= \frac{1}{2}$.

(Ⅰ) Find the values of $a$ and $b$;

(Ⅱ) Let $g(x)=x^{2}-2mx+m$, if for any $x_{1} \in \left[ \frac{1}{2},2\right]$, there always exists $x_{2} \in \left[ \frac{1}{2},2\right]$ such that $g(x_{1})\geqslant f(x_{2})-\ln x_{2}$, find the range of the real number $m$.

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