Challenge Your Brain 3

Geometry Level 4

Let \(ABCDEF\) be a regular hexagon. Let G, H, I, J, K, and L be the midpoints of sides AB, BC, CD, DE, EF, and AF, respectively. The segments \(\overline{AH}\), \(\overline{BI}\), \(\overline{CJ}\), \(\overline{DK}\), \(\overline{EL}\), and \(\overline{FG}\) bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of \(ABCDEF\) be expressed as a fraction \(\frac{m}{n}\) where m and n are relatively prime positive integers. Find m + n.

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