
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
A circle with center at point \( O \) lies on the hypotenuse \( AC \) of the right triangle \( ABC \) and touches its legs \( AB \) and \( BC \). Find \( AC \), given that \( AM = \frac{20}{9} \) and \( AN: MN = 6:1 \), where \( M \) is the point of tangency of \( AB \) with the circle, and \( N \) is the point of intersection of the circle with \( AC \), located between points \( A \) and \( O \).

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