
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 function $f(x)=2\cos ^{2}x+2 \sqrt {3}\sin x\cos x+a$, and the minimum value of $f(x)$ is 2 when $x\in[0, \frac {π}{2}]$.
(1) Find the value of $a$ and the interval where $f(x)$ is monotonically increasing;
(2) First, the horizontal coordinate of the points on the graph of the function $y=f(x)$ is reduced to half of the original, and then the graph is shifted to the right by $\frac {π}{12}$ units to obtain the graph of the function $y=g(x)$. Find the sum of all roots of the equation $g(x)=4$ in the interval $[0, \frac {π}{2}]$.

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