
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 that the complex numbers \( a_{1}, a_{2}, a_{3} \) satisfy
\[
\begin{aligned}
& a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = a_{1}^{3} + a_{2}^{3} + a_{3}^{3} \\
& = a_{1}^{4} + a_{2}^{4} + a_{3}^{4} = 0,
\end{aligned}
\]
find all possible values of \( a_{1} + a_{2} + a_{3} \).

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