
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 that \( x, y, z \) are positive real numbers such that \( x + y + z = 1 \), find the minimum value of the function \( f(x, y, z) = \frac{3x^{2} - x}{1 + x^{2}} + \frac{3y^{2} - y}{1 + y^{2}} + \frac{3z^{2} - z}{1 + z^{2}} \), and provide a proof.

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