Double Trouble

Geometry Level 5

The alternate (interior) angles of a regular pentagon are joined to form another regular pentagon. The ratio of the area of the smaller pentagon to the area of the bigger pentagon in simplest form is \( ( a - \sqrt{b} \times c ) : d,\) where \( a ,b ,c , d \) all are pairwise co-prime and \(a>0.\) Find \( a + b + c + d.\)

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