10. Define the "coordinate distance" between two points $P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right)$ as: $d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$. If the "coordinate distance" from $C(x, y)$ to points $A(1,3), B(6,9)$ is equal, where the real numbers $x, y$ satisfy $0 \leqslant x \leqslant 10,0 \leqslant y \leqslant 10$, then the sum of the lengths of all possible trajectories of point $C$ is $\qquad$.
Solve:
$$
|x-1|+|y-3|=|x-6|+|y-9| \Rightarrow\left\{\begin{array}{l}
y=\frac{17}{2}, x<1,3 \leqslant y \leqslant 9, \\
y=-x+\frac{19}{2}, 1 \leqslant x<6,3 \leqslant y \leqslant 9, \\
y=\frac{7}{2}, x \geqslant 6,3 \leqslant y \leqslant 9
\end{array}\right.
$$

The sum of the lengths of the trajectory of point $C$ that satisfies the conditions is $5+5 \sqrt{2}$.