
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 \( \triangle ABC \) with \( AB = 15 \), \( BC = 14 \), and \( AC = 16 \), let \( N \) be a point on \( \overline{AC} \) such that the circumcircles of \( \triangle ABN \) and \( \triangle BNC \) have equal radii. If \( AN = x \) and \( NC = 16 - x \), determine \( \frac{x}{16-x} \).

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