
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 $10$. There are four distinct points $D_1, D_2, D_3, D_4$ on line $AB$ such that $AD_1 = 3$, $AD_2 = 4$, $AD_3 = 6$, and $AD_4 = 8$. There are points $E_1, E_2, E_3, E_4$ such that each of the triangles $AD_iE_i$ is congruent to $ABC$. Compute the sum $\sum_{k=1}^4 (CE_k)^2$.

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