
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
Let $a$, $b$, and $c$ be nonzero real numbers such that $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5=a^7+b^7+c^7$. Find the value of $a^2+b^2+c^2$.

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