
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 $f(x)=2 \sqrt {3}\cos ^{2}ωx+2\sin ωx\cos ωx- \sqrt {3}\ (ω > 0)$, the distance between two adjacent highest points on its graph is $\dfrac {2}{3}π$.
(I) Find the value of $ω$;
(II) Translate the graph of the function $y=f(x)$ to the right by $\dfrac {π}{6}$ units, then stretch the horizontal coordinate of each point on the resulting graph to twice its original length (while keeping the vertical coordinate unchanged) to obtain the graph of $y=g(x)$. Find the interval(s) on which $g(x)$ is increasing on $[0, \dfrac {4π}{3}]$;
(III) Under the conditions of (II), find the sum of all real roots of the equation $g(x)=t (0 < t < 2)$ in the interval $[0, \dfrac {8}{3}π]$.

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