In the above diagram, square \(ABCD\) is inscribed in circle \(O,\) which is in turn inscribed in square \(A'B'C'D'.\) If the length of \(\overline{A'D'}\) is \(\lvert\overline{A'D'}\rvert=8,\) what is \(\lvert\overline{AD}\rvert?\)

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In the above diagram, \(\square ABCD\) is a square inscribed in circle \(O,\) and \(\overline{ON}\) is perpendicular to \(\overline{BC}.\) If the length of \(\overline{MN}\) is \(2\sqrt{2}-2,\) what is the length of \(\overline{BM}?\)

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