
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
Equilateral triangle $PQR$ has a side length of $10$. There are four distinct triangles $PU_1V_1$, $PU_1V_2$, $PU_2V_3$, and $PU_2V_4$, each congruent to triangle $PQR$, with $QU_1 = QU_2 = 3$. Determine $\sum_{k=1}^4(PV_k)^2$.

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