
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
If \(x=\frac{1+i\sqrt{3}}{2}\) and \(y=\frac{1-i\sqrt{3}}{2}\), where \(i^2=-1\), determine which of the following is true:
\(x^4+y^4\), \(x^6+y^6\), \(x^8+y^8\), and \(x^{10}+y^{10}\).
A) $\ x^4+y^4 = 0$
B) $\ x^6+y^6 = 2$
C) $\ x^8+y^8 = 0$
D) $\ x^{10}+y^{10} = 2$

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