
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
Distinct real numbers \(x, y, z\) are such that among the three numbers

\[ 
\frac{x+y}{x^{2}+xy+y^{2}}, \quad \frac{y+z}{y^{2}+yz+z^{2}}, \quad \frac{z+x}{z^{2}+zx+x^{2}} 
\]

some two are equal. Is it true that all three of these numbers are equal?

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