In a solar system of $n$ planets, Zorn the World Conqueror can invade $m$ planets at a time, but once there are less than $m$ free worlds left, he stops. If he invades $13$ at a time then there are $6$ left, and if he invades $14$ at a time then there are $5$ left. If this solar system has more than $100$ planets, what is the smallest number of planets it could have?
The given information translates to the congruences \begin{align*}
&n\equiv 6\pmod{13},\\
&n\equiv 5\pmod{14}.
\end{align*}From the first congruence we obtain that $n = 6 + 13k$ for some integer $k.$ Combining this result with the second congruence, we have $6+13k = n \equiv 5 \pmod {14}.$ Therefore, $k \equiv 1 \pmod {14}.$ So, $k = 1+14t$ for some integer $t.$ Substituting $1+14t$ for $k,$ we have  \begin{align*}
n&=6+13k \\
&=6+13(1+14t) \\
&=19+182t\equiv 19\pmod{182}.
\end{align*}The smallest such $n$ greater than $100$ is $\boxed{201}$.