
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
The "coordinate distance" between two 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 coordinate distance from point \( C(x, y) \) to point \( A(1,3) \) is equal to the coordinate distance from \( C(x, y) \) to point \( B(6,9) \), where the real numbers \( x \) and \( y \) satisfy \( 0 \leqslant x \leqslant 10 \) and \( 0 \leqslant y \leqslant 10 \), then what is the total length of the trajectory of all points \( C \) that satisfy these conditions?

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