
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 $ABC$ has a side length of $\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\angle ABD_1 = 30^\circ$ and $\angle ABD_2 = 150^\circ$. Determine the sum $\sum_{k=1}^4 (CE_k)^2$.

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