
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
Triangle $ABC$ has sides $AB = 15$, $BC = 20$, and $AC = 25$. Divide $\overline {AB}$ into 60 congruent segments with points $A = P_0, P_1, \ldots, P_{60} = B$, and $\overline {BC}$ into 60 congruent segments with points $B = Q_0, Q_1, \ldots, Q_{60} = C$. For $1 \le k \le 59$, draw segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AC}$ with points $A = R_0, R_1, \ldots, R_{60} = C$. Draw the altitude from $A$ to $\overline {BC}$ at point $D$. Find the sum of the lengths of the 119 segments drawn.

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