Let three vectors $\vec a=\alpha\hat i+4\hat j+2\hat k,;\vec b=5\hat i+3\hat j+4\hat k,;\vec c=x\hat i+y\hat j+z\hat k$ form a triangle such that $\vec c=\vec a-\vec b$ and the area of the triangle is $5\sqrt{6}$. If $\alpha$ is a positive real number, then $\lvert\vec c\rvert$ is equal to: