I wish to solve exactly this formula involving sums and products ⟨ n l ⟩...

I wish to solve exactly this formula involving sums and products
$$\begin{gathered} ⟨n_l⟩={[1+\sum _{k\ne l}\left(e^{bN(l-k)}\frac{\prod _{j\ne l}(1-e^{b(l-j)})}{\prod _{j\ne k}(1-e^{b(k-j)})}\right)]}^{-1}(N+\sum _{h\ne l}\frac{1}{1-e^{b(h-l)}})- \\ \sum _{h\ne l}\left\{{[1+\sum _{k\ne h}\left(e^{bN(h-k)}\frac{\prod _{j\ne h}(1-e^{b(h-j)})}{\prod _{j\ne k}(1-e^{b(k-j)})}\right)]}^{-1}\left(\frac{1}{1-e^{b(l-h)}}\right)\right\} \end{gathered}$$
where b is a positive real number, N is a natural number, and all the sums and products are understood to run from 0 to $M-1$, where M is yet another natural number (all the l, h, j, k indices are therefore bound to this interval, with the appropriate restrictions written under the sums and the products). Eventually I would also like to take the limit as M goes to infinity, but I'm sure how well I can do that.
Do you think there is a hope to massage this formula to make it a little less ugly? For the moment, I've been trying to solve it numerically...