
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  $a,b,c$  be positive real numbers. Find all real solutions  $(x,y,z)$  of the system:
\[ ax+by=(x-y)^{2}
 by+cz=(y-z)^{2}
 cz+ax=(z-x)^{2}\]

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