
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 $a=(m,\cos 2x)$ and $b=(\sin 2x,n)$, and the function $f(x)=a\cdot b$, whose graph passes through the points $(\frac{π}{12}, \sqrt{3})$ and $(\frac{2π}{3},-2)$.

(1) Find the values of $m$ and $n$;
(2) After translating the graph of $y=f(x)$ to the left by $φ(0 < φ < π)$ units to obtain the graph of $y=g(x)$, 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 monotonically increasing.

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