Calculus Level 5

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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