7. Given a moving point $P$ on the $x$-axis, $M, N$ are on the circles $(x-1)^{2}+(y-2)^{2}=1$ and $(x-3)^{2}+(y-4)^{2}=3$ respectively, then the minimum value of $|P M|+|P N|$ is $\qquad$ .
Let $C_{1}(1,2), C_{2}(3,4)$, then the minimum value of $\left|P C_{1}\right|+\left|P C_{2}\right|$ is $\left|C_{1}^{\prime} C_{2}\right|=\sqrt{(3-1)^{2}+(4+2)^{2}}=2 \sqrt{10}$. Therefore, the minimum value of $|P M|+|P N|$ is $2 \sqrt{10}-\sqrt{3}-1$.