Find a uniform lower bound

Let $W=\{w_k: 1\le k\le N\}$ be sequence of nonzero, distinct, real numbers with $\sum\limits_{k\ge 1}\frac{1}{|w_k|}<\infty$ and $\xi_0$ be a fixed number in $(0,1)$. Find the uniform bound of $$f_N=\prod_{k=1}^{N}(1+e^{2\pi w_k})\sum_{k=1}^N a_{k,N}\frac{e^{2\pi \xi_0 w_k}}{1+e^{2\pi w_k}},$$ where $a_{k,N}=w_k\prod\limits_{j=1, j\ne k}\frac{w_j}{w_j-w_k}$. Can we find a finite positive number $C$ such that $$|f_N|\ge C>0,\text{ uniformly in }N.$$ I have tried it as follows. $f_N=P_NS_N$, where $P_N=\prod_{k=1}^{N}(1+e^{2\pi w_k})$ and $S_N=\sum_{k=1}^N a_{k,N}\frac{e^{2\pi \xi_0 w_k}}{1+e^{2\pi w_k}}$. Here $$S_N=\sum_{k=1}^{N} h(w_k)L_k(0),$$ where $L_k(x)=\prod\limits_{j=1, j\ne k}\frac{x-w_j}{w_k-w_j}$ is Lagrange basis function and $h(w)=w\frac{e^{2\pi \xi_0 w}}{1+e^{2\pi w}}$. If $w_k$ tends to $\infty$, then $Q_N$ blows up and $S_N$ decays rapidly, their product $|f_N|>C_1>0$ (verified numerically for large N using MATLAB). So maybe we cannot go separately. Moreover, I got $$\frac{1}{2\pi i}\int_{C_N} H_N(z)dz=-S_N+\psi(0).$$ where $\psi(z)=\frac{e^{2\pi \xi_0 z}}{1+e^{2\pi z}}$, $H_N(z)=\frac{\psi(z)}{zG_N(z)}$, $G_N(z)=\prod\limits_{j=1}^N(1-\frac{z}{w_j})$, and $C_N$ is a rectangle contour with $|y|<\frac{1}{2}$ contains poles $w_1,\cdots, w_N$. Since $\psi(z)$ has pole has at $z=(m+\frac{1}{2})i,~m\in\mathbb{Z}$, $C_N$ contains no poles on imaginary axis.