
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
Rectangle $ABCD$ has side lengths $\overline{AB} = 6$ and $\overline{CB} = 8$. Divide $\overline{AB}$ into 120 congruent segments with points $A=P_0, P_1, \ldots, P_{120}=B$, and divide $\overline{CB}$ into 120 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{120}=B$. For $1 \le k \le 119$, draw segments $\overline{P_kQ_k}$. Repeat this construction on the sides $\overline{AD}$ and $\overline{CD}$, then draw the diagonal $\overline{AC}$. Find the sum of the lengths of the 239 parallel segments drawn.

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