HMMT November 2022 E4

Let $x<0.1$ be a positive real number. Let the foury series be $4+4 x+4 x^{2}+4 x^{3}+\ldots$, and let the fourier series be $4+44 x+444 x^{2}+4444 x^{3}+\ldots$ Suppose that the sum of the fourier series is four times the sum of the foury series. Compute $x$.