
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 \( O \) is the circumcenter of an acute-angled triangle \( ABC \), and \( H \) is the orthocenter of this triangle. It turns out that line \( OH \) is parallel to side \( BC \). On the plane, a point \( K \) is marked such that \( ABHK \) is a parallelogram. Segments \( OK \) and \( AC \) intersect at point \( L \). In what ratio does the perpendicular dropped from point \( L \) to segment \( AH \) divide \( AH \)?

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