
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
We say a nondegenerate triangle whose angles have measures  $\theta_1$ ,  $\theta_2$ ,  $\theta_3$  is *quirky* if there exists 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  $n\ge 3$  for which a triangle with side lengths  $n-1,n,n+1$  is quirky.

*Evan Chen and Danielle Wang*

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