
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 \( x \) and \( y \) be two non-zero numbers such that \( x^2 + xy + y^2 = 0 \) (where \( x \) and \( y \) are complex numbers, but that is not too important). Find the value of

\[ \left( \frac{x}{x+y} \right)^{2013} + \left( \frac{y}{x+y} \right)^{2013} \]

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