Radicals + Sines = Headache

Geometry Level 5

\[ \large 4\sin { 27^\circ } =\sqrt { a+\sqrt { b } } -\sqrt { c-\sqrt { d } } \]

Given that \(a,b,c\) and \(d\) are positive integers that satisfy the equation above with \(b\) and \(d\) are square free and at least \(2\) of \(a,b,c,d\) are equal, find \(a+b+c+d\).

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