
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 side length \( \sqrt{123} \). There are four distinct triangles \( AD_1E_1 \), \( AD_1E_2 \), \( AD_2E_3 \), and \( AD_2E_4 \), each congruent to triangle \( ABC \), with \( BD_1 = BD_2 = \sqrt{13} \). Find \( \sum_{k=1}^4(CE_k)^2 \).

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