
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 $\overrightarrow{m}=(b\sin x,a\cos x)$, $\overrightarrow{n}=(\cos x,-\cos x)$, and $f(x)=\overrightarrow{m}\cdot\overrightarrow{n}+a$, where $a$, $b$, and $x\in R$. Also, $f(\frac{\pi}{6})=2$ and $f'(0)=2\sqrt{3}$.

(I) Find the values of $a$ and $b$.
(II) If the equation $f(x)-\log_{\frac{1}{3}}k=0$ always has real solutions in the interval $[0,\frac{2\pi}{3}]$, find the range of real number $k$.

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