Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let a be the middle term in the expansion of $\left(2+\frac{1}{\sqrt{2}}\right)^{200}$. If $\frac{{ }^{200} \mathrm{C}_{99} \mathrm{~K}}{\mathrm{a}}=\frac{2^{\prime} \mathrm{m}}{\mathrm{n}}$ where m and n are odd numbers, then the ordered pair $(\ell, n)$ is equal to :