
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
Compute the number of ordered quadruples of complex numbers  $(a,b,c,d)$  such that
\[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \]
holds for all complex numbers  $x, y$ .

*Proposed by Evan Chen*

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