
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 \( \overline{FH} \) of length 7. Divide \( \overline{EF} \) into 100 congruent segments with points \( E=P_0, P_1, \ldots, P_{100}=F \), and divide \( \overline{FH} \) into 100 congruent segments with points \( F=Q_0, Q_1, \ldots, Q_{100}=H \). For \( 1 \le k \le 99 \), draw the segments \( \overline{P_kQ_k} \). Repeat this construction on the sides \( \overline{EG} \) and \( \overline{HG} \), and then draw the diagonal \( \overline{EH} \). Find the sum of the lengths of the 199 parallel segments drawn.

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