
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
The distance between the centers \( O_{1} \) and \( O_{2} \) of the circles \( \omega_{1} \) and \( \omega_{2} \) is \( 10r \), and their radii are \( 5r \) and \( 6r \) respectively. A line intersects circle \( \omega_{1} \) at points \( M \) and \( N \) and is tangent to circle \( \omega_{2} \) at point \( K \), with \( MN = 2NK \). Find the length of the chord \( MN \).

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