
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 segment \( AB \) of length 10, a circle \(\omega\) is constructed as its diameter. A tangent line to \(\omega\) is drawn through point \( A \), and a point \( K \) is chosen on this tangent. A line passing through point \( K \), different from \( AK \), touches the circle \(\omega\) at point \( C \). The altitude \( CH \) of triangle \( ABC \) intersects 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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