
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
Given an equilateral triangle \(ABC\). On the extensions of its sides \(AB\) and \(AC\) beyond points \(B\) and \(C\), points \(D\) and \(E\) are taken such that \(|BD| \cdot |CE| = |BC|^2\). Find the locus of points of intersection of the lines \(DC\) and \(BE\).

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