The rays $l_1,l_2,\ldots,l_{n-1}$ divide a given angle $ABC$ into $n$ equal parts. A line $l$ intersects $AB$ at $A_1$, $BC$ at $A_{n+1}$, and $l_i$ at $A_{i+1}$ for $i=1,\ldots,n-1$. Show that the quantity $$\left(\frac1{BA_1}+\frac1{BA_{n+1}}\right)\left(\frac1{BA_1}+\frac1{BA_2}+\ldots+\frac1{BA_{n+1}}\right)^{-1}$$is independent of the line $l$, and compute its value if $\angle ABC=\phi$.