7. Given moving points $P$, $M$, and $N$ are on the $x$-axis, circle $\Gamma_{1}$: $(x-1)^{2}+(y-2)^{2}=1$, and circle $\Gamma_{2}$: $(x-3)^{2}+$ $(y-4)^{2}=3$, respectively. Then the minimum value of $|P M|+|P N|$ is $\qquad$
7. $2 \sqrt{10}-\sqrt{3}-1$

It is known that the center coordinates of circle $\Gamma_{1}$ are $(1,2)$, and the center coordinates of the circle $\Gamma_{2}$ symmetric to the $x$-axis are $(3,-4)$.
Then the minimum value of $|P M|+|P N|$ is
$$
\begin{array}{l}
\sqrt{(3-1)^{2}+(-4-2)^{2}}-1-\sqrt{3} \\
=2 \sqrt{10}-\sqrt{3}-1
\end{array}
$$