7. Let $m, n, p, l$ be natural numbers. If the following system:

$$
\left\{\begin{array}{l}
n p^{2}+m l^{2} \leq 23456 \\
m n+p^{7} l=1626 \\
m(l+1)^{p}+n p l=2197+m
\end{array}\right.
$$

has solutions, then present the one for which the sum $m+n+p+l$ is minimal. Otherwise, prove that there are no solutions.
Solution: $m(l+1)^{p}-m=m\left((l+1)^{p}-1\right) \vdots l$, then from the last line of the system we get that $2197=13^{3}: l$, from which $l=1,13,13^{2}$ or $13^{3}$. From the first line of the system $l^{2} \leq m l^{2} \leq 23456$, from which $l=1$ or $l=13$. Further, $l p^{7} \leq 1626$, from which $p=1,2$ when $l=1$ and $p=1$ when $l=13$. It remains to consider three cases: $(l, p)=(1,1),(1,2),(13,1)$. In each case, the second and third equations of the system give relations on $m, n$, which lead to a quadratic equation without integer solutions. Answer: no solutions.