
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 real number $\lambda$ such that for any three complex numbers $\mathrm{z}_{1}$, $\mathrm{z}_{2}$, $\mathrm{z}_{3} \in \{\mathrm{z} \in \mathbb{C} \mid |\mathrm{z}| < 1\}$, if $\mathrm{z}_{1} + \mathrm{z}_{2} + \mathrm{z}_{3} = 0$, then $\left| \mathrm{z}_{1} \mathrm{z}_{2} + \mathrm{z}_{2} \mathrm{z}_{3} + \mathrm{z}_{3} \mathrm{z}_{1} \right|^{2} + \left| \mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}_{3} \right|^{2} < \lambda$.

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