
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 vectors $\overrightarrow{m}=(1,\sqrt{3})$, $\overrightarrow{n}=(\sin x,\cos x)$, let function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$

(I) Find the smallest positive period and maximum value of function $f(x)$;

(II) In acute triangle $\Delta ABC$, let the sides opposite angles $A$, $B$, $C$ be $a$, $b$, $c$ respectively. If $c=\sqrt{6}$, $\cos B=\frac{1}{3}$, and $f(C)=\sqrt{3}$, find $b$.

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