
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
A real number  $\alpha \geq 0$  is given. Find the smallest  $\lambda  = \lambda (\alpha ) > 0$ ,  such that for any complex numbers  ${z_1},{z_2}$  and  $0 \leq x \leq 1$ ,  if   $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$ ,  then   $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$ .

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