
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]


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In the triangle shown, \( n \) is a positive integer, and \( \angle A < \angle B < \angle C \). The side lengths of the triangle are labeled as follows: \( AB = 2n + 10 \), \( AC = 3n - 6 \), \( BC = 2n + 16 \). Determine how many possible values of \( n \) there are.

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