
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 $\overrightarrow{a}=(m,\cos 2x)$ and $\overrightarrow{b}=(\sin 2x,n)$, let the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, and the graph of $y=f(x)$ passes through the points $\left( \frac {\pi}{12}, \sqrt {3}\right)$ and $\left( \frac {2\pi}{3},-2\right)$.  
(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 from each highest point on the graph of $y=g(x)$ to the point $(0,3)$ is $1$, find the intervals of increase for $y=g(x)$.

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