
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
A unit circle is centered at  $(0, 0)$  on the  $(x, y)$  plane. A regular hexagon passing through  $(1, 0)$  is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as 
    \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \]
    for positive integers  $m$ ,  $n$ ,  $p$ , and  $q$  such that  $m$  and  $n$  are relatively prime and  $p$  is squarefree. Find  $m+n+p+q$ .

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