2021 AMC 12B Problem 21

Let S be the sum of all positive real numbers x for whichx^{2^{\sqrt2}}=\sqrt2^{2^x}.Which of the following statements is true?

\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.