The number $$\sqrt{104\sqrt6+468\sqrt{10}+144\sqrt{15}+2006}$$can be written as $a\sqrt2+b\sqrt3+c\sqrt5$, where $a$, $b$, and $c$ are positive integers.  Find $a\cdot b\cdot c.$

Let's think step by step and output the final answer within \boxed{}.
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