
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
The area of triangle \( ABC \) is 1, with \( AC = 2BC \). Point \( K \) is the midpoint of side \( AC \). A circle centered at \( K \) intersects side \( AB \) at points \( M \) and \( N \) such that \( AM = MN = NB \). Find the area of the part of triangle \( ABC \) that is enclosed within the circle.

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