
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
Let \( x, y, z, A, B, C \in \mathbb{R} \), with
\[ A + B + C = \pi. \]

Define \( G_n = x^n \sin nA + y^n \sin nB + z^n \sin nC \) for \( n \in \mathbb{Z}_{+} \).

Given \( G_1 = G_2 = 0 \), find the value of \( G_n \).

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