Looks like an spherical square in the middle

Geometry Level 5

A disc of radius 1 unit is cut into 4 quadrants. These are placed in a square of side 1 unit (these quadrants do not have any part outside the square). What is the least possible area of overlap shared by all quadrants?

If this area can be written as \(\dfrac {a+b\sqrt{c}+d\pi}{e}\), where \(a\), \(b\), \(c\) \(d\) and \(e\) are integers, with \(c\) and \(e\) being positive, \(c\) being square-free and \(\gcd(a,b,d,e)=1\), find \(abcde\).

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