How Long Can The Product Go!!!

Geometry Level pending

Suppose a $$3\text{D}$$ vector $$\vec{v}=(a,b,c)$$ with $$a,b,c\geq 0$$ makes angles $$\alpha,\beta,\gamma$$ with the $$x,y,z$$ axes respectively. The direction cosines of $$\vec{v}$$ is then defined by the followings: $\cos \alpha=\dfrac{a}{\sqrt{a^2+b^2+c^2}}$ $\cos\beta=\dfrac{b}{\sqrt{a^2+b^2+c^2}}$ $\cos \gamma=\dfrac{c}{\sqrt{a^2+b^2+c^2}}$ The maximum value of $$\cos\alpha \cos\beta\cos \gamma$$ can be expressed as $$\dfrac{\sqrt{m}}{n}$$ where $$(m,n)\in\mathbb{N}^2$$. Find the value of $$m+n$$.

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