
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 passes through the vertices \( A \) and \( N \) of triangle \( A C N \) and intersects its sides \( A C \) and \( C N \) at points \( B \) and \( K \), respectively, other than the vertices of the triangle. The ratio of the area of triangle \( B C K \) to the area of triangle \( A C N \) is \( \frac{1}{4} \).

a) Find the ratio \( A N: B K \).

b) Additionally, it is known that the ratio of the areas of triangles \( B C N \) and \( A C K \) is \( \frac{9}{16} \). Find the ratio \( N K: A B \).

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