AMC 12 2025a Problem 7

In a certain alien world, the maximum running speed $v$ of an organism is dependent on its number of toes $n$ and number of eyes $m$. The relationship can be expressed as $v=kn^am^b$ centimeters per hour, where $k$, $a$, and $b$ are integer constants. In a population where all organisms have $5$ toes, $\log v=4+2\log m$; and in a population where all organisms have $25$ eyes, $\log v=4+4\log n$, where the logarithms are base 10. What is $k+a+b$?

\textbf{(A)}~20 \textbf{(B)}~21 \textbf{(C)}~22 \textbf{(D)}~23 \textbf{(E)}~24