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.

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.