
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 triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) intersect at point \( K \), find the value of \( \frac{|KO|}{|KH|} \).

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