Showing that $\left|\prod_{k=1}^\infty \cos\left(\frac{2\pi}{3^k}\right)\right| > 0$

Question

I was wondering what would be an easy argument to show that the infinite product $\prod_{k=1}^\infty \cos\left(\frac{2\pi}{3^k}\right)$ does not converge to zero? I understand that the important part is the fact that already after three terms $\cos\left(\frac{2\pi}{3^k}\right)\approx 1$ and $\cos\left(\frac{2\pi}{3^k}\right)\to 1$ quite fast as $k\to \infty$. But I am not sure how to transform this into a rigorous argument. One can observe via Desmos etc. that the product seems to converge to $\approx 0.371$.