
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 \(a, b, c, \lambda, \mu \in \mathbf{R}_{+}\), and \(a + b + c = 1\). For any positive integers \(m, n \geq 2\), the following inequality holds:
$$
\frac{a^{m}+b^{n}}{\lambda b+\mu c}+\frac{b^{m}+c^{n}}{\lambda c+\mu a}+\frac{c^{m}+a^{n}}{\lambda a+\mu b} 
\geq \frac{3^{2-m}+3^{2-n}}{\lambda+\mu} .
$$

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