
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 a rectangle $XYZW$ with sides $\overline{XY} = 5$ and $\overline{YZ} = 6$, divide $\overline{XY}$ into 210 congruent segments with points $X=P_0, P_1, \ldots, P_{210}=Y$, and divide $\overline{YZ}$ into 210 congruent segments with points $Y=Q_0, Q_1, \ldots, Q_{210}=Z$. For $1 \le k \le 209$, draw the segments $\overline{P_kQ_k}$. Repeat this construction on the sides $\overline{XZ}$ and $\overline{WZ}$, and then draw the diagonal $\overline{XZ}$. Find the sum of the lengths of the 419 parallel segments drawn.

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