
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 $a=(2\cos x, \sqrt{3}\sin x)$ and $b=(\cos x,2\cos x)$, and the function $f(x)=a\cdot b+m(m\in R)$, where $x\in\left[ 0, \frac{\pi}{2} \right]$ and the minimum value of $f(x)$ is $2$.  
$(1)$ Find the interval of monotonic increase for $f(x)$.  
$(2)$ First, keep the y-coordinates of all points on the graph of $y=f(x)$ unchanged and reduce the x-coordinates to half of their original values, then shift the resulting graph to the right by $\frac{\pi}{12}$ units to obtain the graph of $y=g(x)$. Find the sum of all roots of the equation $g(x)=4$ in the interval $\left[ 0, \frac{\pi}{2} \right]$.

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