3. In the Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where the real numbers $x$ and $y$ satisfy $0 \leqslant x \leqslant 10$ and $0 \leqslant y \leqslant 10$, then the sum of the lengths of all the loci of points $C$ that satisfy the condition is . $\qquad$
3. $5(\sqrt{2}+1)$ Detailed Explanation: From the condition, we get $|x-1|+|y-3|=|x-6|+|y-9|$. (1)

When $y \leqslant 3$, (1) becomes $|x-1|=6+|x-6|$, no solution;
When $y \geqslant 9$, (1) becomes $|x-1|+6=|x-6|$, no solution;
When $3 \leqslant y \leqslant 9$, (1) becomes $2 y-12=|x-6|-|x-1|$.
If $x \leqslant 1$, then $y=8.5$, the length of the line segment is 1; if $1 \leqslant x \leqslant 6$, then $x+y=9.5$, the length of the line segment is $5 \sqrt{2}$; if $x \geqslant 6$, then $y=3.5$, the length of the line segment is 4. In summary, the sum of the lengths of the line segments that form the trajectory of point $C$ is $1+5 \sqrt{2}+4=5(\sqrt{2}+1)$.