calculation puzzle - Paint the rectangle with least movement

There is a painting brush tool that can paint a circle area of $1$ unit radius. What is the shortest paint track for painting at least a $100*100$ units rectangle area on wall? How much did the brush moved in total?

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Rules:

• Brush must touch wall only once. (It says that painting should be continues and you have only one shot)


• Painting can start and finish anywhere.


• Painting outside the rectangle does not matter. (Actually it is inevitable to paint some of the outer area of rectangle)

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Example:

Here's a naive try for a $10*10$ rectangle and the same brush with radius of $1$ unit result is $58$ units for $10*10$ rectangle and $(n*m)+2m-2$ for $n*2m$ rectangle

Answer

The best I can do (so far) is a total track length of 5083.592 (see below).

Update: You can also do away with the final returning curved path in my final solution, which chops off pi/4, so the total length is reduced to 5082.807