There exists \(2\) points on Cartesian coordinate. Point \(B\) has coordinates \((0,1)\). Point \(A\) is on the origin \((0,0)\).

Point \(A\) and \(B\) always has a constant distance of \(1\) from each other.

Point \(B\) moves with a horizontal velocity (parallel to the x-axis) of \(v.\)

Point \(A\) moves with a horizontal velocity of \(2v.\)

Point \(A\)'s vertical velocity (parallel to the y-axis) is \(0\) while Point \(B\) is allowed to move vertically in order to keep the constant distance of \(1\).

All this movement is happening in the first quadrant.

Let \(P\) be the area made by the figure defined by: the \(x\) axis, the \(y\) axis, and the path traveled by point \(B\) up to where it meets the \(x\) axis. Find \[\left\lfloor 1000P \right\rfloor .\]

After you solve this, you might want to try a continuation of this problem.

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