Its not a triangle. It's not a sector. It's a Trianglector!

Geometry Level 4

Square \(ABCD\) has side length \(1\). The quarter-circle sectors \(BAC\) and \(ABD\) are drawn, along with diagonal \(AC\). If the area of the shaded region is \(\frac{a \pi + b \sqrt{c} + d}{e}\), where \(a,b,c,d,e\) are all integers with \(c\) not divisible by the square of any prime and \(\text{gcd}(a,e) = 1\), find \(a^2 + b^2 + c^2 + d^2 + e^2\).

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