You have the 3 numbers \(\big\{4,\ 1+2\sqrt{2},\ \sqrt{2}\big\}\) on a blackboard. You are permitted to perform the following operation whenever you want and as many times as you want:

- You can choose 2 numbers \(a\) and \(b\) and replace them with \(\frac{a+b}{\sqrt{2}}\) and \(\frac{a-b}{\sqrt{2}}\).

Is it possible to attain the 3 numbers \(\big\{3+\sqrt{2},\ 2\sqrt{2},\ \sqrt{2}-1\big\}\) after a finite number of operations?

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