Let $\zeta = e^{2\pi i/99}$ and $\omega = e^{2\pi i/101}$. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \cdots + a_1x + a_0 \] has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.

Let's think step by step and output the final answer within \boxed{}.
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