
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]


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Given vectors $\overrightarrow{a}=(m,\cos 2x)$ and $\overrightarrow{b}=(\sin 2x,1)$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, with the graph of $y=f(x)$ passing through the point $\left( \frac {\pi}{12}, \sqrt {3}\right)$.  
$(1)$ Find the value of $m$;  
$(2)$ After shifting the graph of $y=f(x)$ to the left by $\phi(0 < \phi < \pi)$ units to get the graph of the function $y=g(x)$, if the minimum distance from the highest points on the graph of $y=g(x)$ to the point $(0,3)$ is $1$, find the interval of increase for $y=g(x)$.

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