
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
In triangle \( KLM \), a median \( KP \) is drawn. Point \( O \) is the center of the circumcircle of the triangle, and point \( Q \) is the center of the incircle of the triangle. The segments \( KP \) and \( OQ \) intersect at point \( S \) such that \(\frac{OS}{PS} = \sqrt{6} \frac{QS}{KS}\). Find the product of the cosines of the angles \( \angle KLM \) and \( \angle KML \), given that \(\angle LKM = \frac{\pi}{3}\). Round the answer to two decimal places if necessary.

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