
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 EFGH has sides $\overline{EF}$ of length 5 and $\overline{FG}$ of length 2. Divide $\overline{EF}$ into 84 congruent segments with points $E=P_0, P_1, \ldots, P_{84}=F$, and divide $\overline{FG}$ into 84 congruent segments with points $F=Q_0, Q_1, \ldots, Q_{84}=G$. For $1 \le k \le 83$, draw the segments $\overline{P_kQ_k}$. Repeat this construction on the sides $\overline{EH}$ and $\overline{GH}$, and then draw the diagonal $\overline{EG}$. Find the sum of the lengths of the 167 parallel segments drawn.

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