
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 the smallest positive number  $\lambda $  , such that for any complex numbers  ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$  ,if   $z_1+z_2+z_3=0$ , then   $$ \left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda . $$  

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