
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 vectors $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\cos x,3)$.
(I) When $\overrightarrow{m}$ is parallel to $\overrightarrow{n}$, find the value of $\dfrac{\sin x+\cos x}{3\sin x-2\cos x}$;
(II) In acute triangle $ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $\sqrt{3}c=2a\sin(A+B)$. The function is given by $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$. Find the range of $f(B+\frac{\pi}{8})$.

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