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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