Cody has started running in a well-organized manner. He runs \(100 \text{ m}\) east, then turns left and runs another \(10 \text{ m}\) north, turns left and runs \(1 \text{ m},\) again turns left and runs \(0.1 \text{ m},\) and on the next turn \(0.01 \text{ m},\) and so on.

Assuming that Cody can run in this pattern infinitely, the displacement from his initial position can be written as \(\frac{a}{\sqrt{b}}\) with \(a\) and \(b\) being positive integers and \(b\) square-free.

What is the value of \( a \times b?\)

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