The number \(\sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}\) can be written as \(a\sqrt{2}+b\sqrt{3}+c\sqrt{5},\) where \(a, b,\) and \(c\) are positive integers. Find \(a\cdot b\cdot c.\)

(第二十四届AIME1 2006 第5题)