Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.
Source: 2020 Taiwan APMO Preliminary
Tags: Taiwan, algebra, trigonometry
Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.