
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 non-zero real numbers \(a\), \(b\), and \(c\) (which are not all equal) satisfy the equation
$$
\frac{a^{2}}{2 a^{2}+b c}+\frac{b^{2}}{2 b^{2}+c a}+\frac{c^{2}}{2 c^{2}+a b}=1.
$$

Find the value of \(a+b+c\).

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