HMMT 2015 U28

Let $w, x, y$, and $z$ be positive real numbers such that

$$\begin{aligned} 0 & \neq \cos w \cos x \cos y \cos z \\ 2 \pi & =w+x+y+z \\ 3 \tan w & =k(1+\sec w) \\ 4 \tan x & =k(1+\sec x) \\ 5 \tan y & =k(1+\sec y) \\ 6 \tan z & =k(1+\sec z) \end{aligned}$$

(Here $\sec t$ denotes $\frac{1}{\cos t}$ when $\cos t \neq 0$.) Find $k$.