
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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We say a nondegenerate triangle whose angles have measures \(\theta_{1}, \theta_{2}, \theta_{3}\) is quirky if there exist integers \(r_{1}, r_{2}, r_{3}\), not all zero, such that
\[ r_{1} \theta_{1} + r_{2} \theta_{2} + r_{3} \theta_{3} = 0. \]

Find all integers \(n \geq 3\) for which a triangle with side lengths \(n-1, n, n+1\) is quirky.

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