
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 vector $\overrightarrow{b}=(\sin 2x,n)$, the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$, and the graph of $y=f(x)$ passes through the points $(\frac{\pi }{12},\sqrt{3})$ and $(\frac{2\pi }{3},-2)$.

(I) Find the values of $m$ and $n$;

(II) After shifting the graph of $y=f(x)$ to the left by $\varphi (0 < \varphi < \pi )$ units, we obtain the graph of the function $y=g(x)$. If the minimum distance between each highest point on the graph of $y=g(x)$ and the point $(0,3)$ is $1$, find the interval where $y=g(x)$ is increasing.

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