AMC 12B 2021 Spring Problem 21

Let S be the sum of all positive real numbers x for which

x^{2^{\sqrt{2}}}=\sqrt{2}^{2^{x}}

Which of the following statements is true?

Answer Choices
A. S<\sqrt{2}
B. S=\sqrt{2}
C. \sqrt{2}<S<2
D. 2 \leq S<6
E. S \geq 6