A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let \(h\) be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then \(h\) can be written in the form \(\frac m{\sqrt{n}},\) where \(m\) and \(n\) are positive integers and \(n\) is not divisible by the square of any prime. Find \(\lfloor m+\sqrt{n}\rfloor.\) (The notation \(\lfloor x\rfloor\) denotes the greatest integer that is less than or equal to \(x.\))
(第二十四届AIME1 2006 第14题)