
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 \(a, b\), and \(c\) are complex numbers satisfying
$$
\begin{aligned}
a^{2}+a b+b^{2} & =1+i \\
b^{2}+b c+c^{2} & =-2 \\
c^{2}+c a+a^{2} & =1,
\end{aligned}
$$
compute \((a b+b c+c a)^{2}\). (Here, \(i=\sqrt{-1}\).)

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