Zig-Zag Midpoint Wag

Geometry Level 5

From vertex \(C\) of a unit equilateral triangle \(ABC,\) we draw a line segment to the midpoint \(m_1\) of the opposite side. From vertex \(m_1\) of triangle \(BCm_1,\) we similarly draw a line segment to the midpoint \(m_2\) of the opposite side. Then we draw line segment \(m_2m_3\) in triangle \(Cm_1m_2,\) and so on. We travel indefinitely in this manner, getting closer and closer to a destination point we call \(P.\)

If \(\overline{AP} + \overline{BP} + \overline{CP}\) can be expressed as \[\dfrac{a}{b}\left(e\sqrt{c} + f\sqrt{d}\right)\] where \(a,b,c,d,e,f\) are positive integers, \(\gcd(a,b) = \gcd(e,f) = 1,\) and \(c\) and \(d\) are distinct and square-free, input \(a + b + c + d + e + f\) as your answer.

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