
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 $\overrightarrow {a}=(2\cos x, \sqrt {3}\sin x)$ and $\overrightarrow {b}=(\cos x, 2\cos x)$, and a function $f(x)= \overrightarrow {a}\cdot \overrightarrow {b}+m$, where $m \in \mathbb{R}$. The minimum value of $f(x)$ when $x \in [0, \frac{\pi}{2}]$ is 2.

1. Find the interval where $f(x)$ is monotonically increasing.
2. First, the vertical coordinates of the graph of $y=f(x)$ remain unchanged, while the horizontal coordinates are reduced to half of the original. Then, the resulting graph is shifted to the right by $\frac{\pi}{12}$ units, yielding the graph of $y=g(x)$. Find the sum of all roots of the equation $g(x)=4$ in the interval $[0, \pi]$.

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