GAIC Math 2024 Problem 67

Given positive number $\alpha, \beta, \gamma, \delta$ satisfying $\alpha+\beta+\gamma+\delta=2\pi$, and $k=\frac{3\tan\alpha}{1+\sec \alpha}=\frac{4\tan \beta}{1+\sec \beta}=\frac{5\tan \gamma}{1+\sec \gamma}=\frac{6\tan\delta}{1+\sec\delta}$, then $k$= _____.