
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
Find all positive real numbers \( k \) such that for any positive real numbers \( a, b, c \), the following inequality holds:
\[ \frac{a}{b+c} + \frac{b}{c+a} + \frac{k c}{a+b} \geq 2. \]

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