Hexagon \(ABCDEF\) is divided into five rhombuses, \(\mathcal{P, Q, R, S,}\) and \(\mathcal{T,}\) as shown. Rhombuses \(\mathcal{P, Q, R,}\) and \(\mathcal{S}\) are congruent, and each has area \(\sqrt{2006}.\) Let \(K\) be the area of rhombus \(\mathcal{T}.\) Given that \(K\) is a positive integer, find the number of possible values for \(K.\)

(第二十四届AIME1 2006 第8题)