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

$(1)$ Find the values of the real numbers $m$ and $n$;

$(2)$ After shifting the graph of $y=f(x)$ to the left by $\phi(0 < \phi < \pi)$ 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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