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)