
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 a triangle \(KLM\) with base \(KM = \frac{\sqrt{3}}{2}\) and side \(KL = 1\). A circle passing through points \(K\) and \(L\) has its center on the altitude \(LF\) dropped to the base \(KM\). It is known that \(FM = \frac{\sqrt{3}}{6}\) and point \(F\) lies on \(KM\). Find the area of the circle determined by this circle.

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