
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 the diagram below, \( ABC \) is a triangle such that \( AB \) is longer than \( AC \). The point \( N \) lies on \( BC \) such that \( AN \) bisects \( \angle BAC \). The point \( G \) is the centroid of \( \triangle ABC \), and it is given that \( GN \) is perpendicular to \( BC \). Suppose \( AC = 6 \text{ cm} \), \( BC = 5 \sqrt{3} \text{ cm} \), and \( AB = x \text{ cm} \). Find the value of \( x \).

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