
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
Given non-zero complex numbers \( x \) and \( y \) satisfying
\[ y^{2}(x^{2}-xy+y^{2})+x^{3}(x-y)=0, \]
find the value of
\[ \sum_{m=0}^{29} \sum_{n=0}^{29} x^{18mn} y^{-18mn}. \]

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