I have a infinitely long strip of paper that is one unit wide. On the paper is a point \(P\) that is \(\dfrac{1}{4}\) units away from one of the edges of the strip.

Let \(L\) be the locus of all points \(O\) such that if I draw the circle with center \(O\) passing through \(P\), the entire circle can be drawn on the strip of paper. If the area of \(L\) can be expressed by \(\dfrac{a\sqrt{b}}{c}\) for relatively prime \(a,c\) and square-free \(b\), then find \(a+b+c\).

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