
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 the side lengths of a triangle and \( x + y + z = 1 \). Find the minimum value of the real number \( \lambda \) such that the inequality
$$
\lambda(x y + y z + z x) \geqslant 3(\lambda + 1)x y z + 1
$$
always holds.

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