
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 side length $\sqrt{44}$. There are two congruent triangles $PT_1U_1$ and $PT_2U_2$, each congruent to triangle $PQR$, with $QT_1 = QT_2 = \sqrt{33}$. Find $\sum_{k=1}^2(PU_k)^2$.

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