Imagine you are staying at the hotel at a point on a plane. There are two infinite perpendicular lines passing through this point, and the rest is regular ground.

You've decided to take a two hour walk, starting and ending at the hotel. On the lines, you walk at a speed of \(\sqrt{3}\) units per hour, and on the ground you walk at \(1\) unit per hour.

Let \(R\) be the set of points that you can reach following the rules. If the area of \(R\) is of the form \(a\sqrt{b}-c\), find \(a+b+c\).

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