
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
Let \( NC = x \), then \( BN = 3x \). Triangle \( ABC \) is isosceles, so \( AB = BC = 4x \). Angle \( ANC \) is a right angle because it is subtended by a diameter. Therefore, angle \( ANB \) is also a right angle since \( ANC \) and \( ANB \) are adjacent.

In the right triangle \( ANB \), using the Pythagorean theorem:

\[ AN^2 = AB^2 - BN^2 = 16x^2 - 9x^2 = 7x^2 \]

In the right triangle \( ANC \), using the Pythagorean theorem:

\[ AC = \sqrt{AN^2 + NC^2} = \sqrt{7x^2 + x^2} = \sqrt{8x^2} = 2\sqrt{2}x \]

Then,

\[ \frac{AC}{BC} = \frac{2\sqrt{2}x}{4x} = \frac{1}{\sqrt{2}}. \]

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