A set of positive numbers has the \(triangle~property\) if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets \(\{4, 5, 6, \ldots, n\}\) of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of \(n\)?

(第十九届AIME2 2001 第5题)