
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
On the segment \( AB \) of length 10, a circle \(\omega\) is constructed with \( AB \) as the diameter. A tangent is drawn to \(\omega\) at point \( A \), on which a point \( K \) is chosen. A line through \( K \) different from \( AK \) is drawn, which touches the circle \(\omega\) at point \( C \). The altitude \(CH\) of triangle \(ABC\) intersects the segment \( BK \) at point \( L \). Find the area of triangle \(CKL\), given that \( BH: AH = 1:4 \).

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