
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 given positive real numbers. Determine all positive real numbers \( x, y \), and \( z \) such that
\[ x + y + z = a + b + c \]
and
\[ 4xyz - (a^2 x + b^2 y + c^2 z) = abc. \]

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