
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
Find the solutions of the system of equations

$$
\begin{aligned}
a^{3}+3 a b^{2}+3 a c^{2}-6 a b c & =1 \\
b^{3}+3 b a^{2}+3 b c^{2}-6 a b c & =1 \\
c^{3}+3 c a^{2}+3 c b^{2}-6 a b c & =1
\end{aligned}
$$

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