
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 triangle shown, $n$ is a positive integer, and $\angle A > \angle C > \angle B$. The side lengths of the triangle are given as $AB = 2n + 15$, $BC = 3n$, and $AC = 2n^2 - 2n + 8$. Determine how many possible values of $n$ are there such that the side lengths form a triangle and the angle conditions are satisfied.

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