Blue vs Red

Geometry Level 3

\(ABCDE\) is a regular pentagon where \(O\) is its center. It is given \(\sin 18^\circ=\dfrac{1}{4}(\sqrt{5}-1)\). The ratio of the blue area to the red area, \(\frac{\color{blue}[\color{blue}B\color{blue}L\color{blue}U\color{blue}E\,\color{blue}]}{{\color{red}[\,\color{red}R}{\color{red}E}{\color{red}D\,\color{red}]}}\), can be denoted as \(\dfrac{\sqrt{a}-b}{c}\), where \(a\), \(b\), and \(c\) are positive integers. Calculate \(a+b+c\).

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