
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
An equilateral triangle $ABC$ has a side length of $3$. There are two distinct triangles $AD_1E_1$ and $AD_2E_2$, each congruent to triangle $ABC$, and two triangles $AF_1E_1$ and $AF_2E_2$, each isosceles with $AF_1 = AF_2 = AE_1 = AE_2 = 2$ and base angles $30^\circ$. Given $BD_1 = BD_2 = 2$, find $\sum_{k=1}^2(CE_k)^2 + (CF_1)^2 + (CF_2)^2$.

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