A right pyramid has a square base with side length 10 meters. The apex of the pyramid is directly above the center of the base, and the slant height of the pyramid is 13 meters. An equilateral triangle is inscribed in the base of the pyramid. If a point is randomly chosen on this equilateral triangle, what is the probability that this point is closer to the apex of the pyramid than to any edge of the equilateral triangle? Express your answer as a simplified fraction \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \). Let's think step by step and output the final answer within \boxed{}.
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