One Bird With The Same Stone... Twice!

Nina throws a stone in such a way that it will hit a bird perched at the top of a tree on its way up. However, once the stone is thrown, the bird flies horizontally at constant velocity in the same plane as the path of the projectile, only to be struck by the stone on its way back down. If the stone's path reaches a maximum height of twice the height of the tree, then the ratio of horizontal velocity of stone to that of the bird is in the form \( \dfrac1n + \dfrac1{\sqrt n} \).

Find \(2n\).

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