
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
Let rectangle \(ABCD\) have sides \(\overline{AB}\) of length 5 and \(\overline{CB}\) of length 4. Divide \(\overline{AB}\) into 200 congruent segments with points \(A=P_0, P_1, \ldots, P_{200}=B\), and divide \(\overline{CB}\) into 200 congruent segments with points \(C=Q_0, Q_1, \ldots, Q_{200}=B\). For \(1 \le k \le 199\), draw the segments \(\overline{P_kQ_k}\). Repeat this construction on the sides \(\overline{AD}\) and \(\overline{CD}\), and then draw the diagonal \(\overline{AC}\). Calculate the total length of all 399 segments drawn.

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