HMMT November 2024 H3

Points $K, A, L, C, I, T, E$ are such that triangles $C A T$ and $E L K$ are equilateral, share a center $I$, and points $E, L, K$ lie on sides $\overline{C A}, \overline{A T}, \overline{T C}$ respectively. If the area of triangle $C A T$ is double the area of triangle $E L K$ and $C I=2$, compute the minimum possible value of $C K$.