
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
Determine the smallest constant \( k > 0 \) such that
$$
\frac{a b}{a+b+2 c}+\frac{b c}{b+c+2 a}+\frac{c a}{c+a+2 b} \leq k(a+b+c),
$$
for all \( a, b, c > 0 \).

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