An equilateral triangle is inscribed in the ellipse whose equation is \(x^2+4y^2=4\). One vertex of the triangle is \((0,1)\), one altitude is contained in the y-axis, and the length of each side is \(\sqrt{\frac mn}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

(第十九届AIME1 2001 第5题)