= Solution
The vertices satisfy
$$
(z+1)^N-1=0.
$$
After removing the root $z=0$, the product of the other roots has <modulus> equal to the constant term of $((z+1)^N-1)/z$, namely $N$. Hence the product of the $N-1$ chord lengths from one vertex of a unit <regular polygon> is $N$. Multiplying this identity over all $N$ vertices counts every chord twice, so the product of all chord lengths is $N^{N/2}$. Scaling the circle by $R$ scales each of its $N(N-1)/2$ chords by $R$, giving
$$
\boxed{N^{N/2}R^{N(N-1)/2}.}
$$
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