
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

Point \( M \) lies on side \( AC \) of an equilateral triangle \( ABC \) with side length \( a \) such that \( AM:MC = 1:2 \). Points \( K \) and \( L \), located on sides \( AB \) and \( BC \) respectively, form the vertices of another equilateral triangle \( MKL \). Find the side lengths of triangle \( MKL \).

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