| $X$ | $k$ | $2 k$ | $4 k$ | Total |
| $\mathrm { P } ( X = x )$ | $\frac { 4 } { 7 }$ | $a$ | $b$ | 1 |
The following is a probability distribution table of the random variable $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$X$ & $k$ & $2 k$ & $4 k$ & Total \\
\hline
$\mathrm { P } ( X = x )$ & $\frac { 4 } { 7 }$ & $a$ & $b$ & 1 \\
\hline
\end{tabular}
\end{center}
$\frac { 4 } { 7 } , a , b$ form a geometric sequence in this order, and the mean of $X$ is 24. Find the value of $k$. [3 points]