
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the Cartesian coordinate system, define the "rectilinear distance" between points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) as \( d(P, Q) = |x_1 - x_2| + |y_1 - y_2| \). If point \( C(x, y) \) has the same "rectilinear distance" to points \( A(1, 3) \) and \( B(6, 9) \), where real numbers \( x \) and \( y \) satisfy \( 0 \leqslant x \leqslant 10 \) and \( 0 \leqslant y \leqslant 10 \), determine the sum of the lengths of the trajectories of all points \( C \) that meet these conditions.

Present the answer in LaTex format: \boxed{Your answer}
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