HMMT 2025 A8

Define $\operatorname {sgn}(x)$ to be 1 when $x$ is positive, $-1$ when $x$ is negative, and 0 when $x$ is 0. Compute

$$\sum_{n = 1}^{\infty}\frac{\operatorname{sgn}(\sin(2^{n}))}{2^{n}}.$$

(The arguments to sin are in radians.)