
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 \( a, b, c, x, y, z \in \mathbf{R}_{+} \) such that:
\[ c y + b z = a, \]
\[ a z + c x = b, \]
\[ b x + a y = c. \]

Find the minimum value of the function
\[ f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}. \]

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