Mathematics Question from JEE Main 2023 · 29 January, Shift 2

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 $\frac{^{200}C_{99}K}{a} = \frac{2^\ell m}{n},$ where $m$ and $n$ are odd numbers, then the ordered pair $(\ell, n)$ is equal to: