
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
For a complex number $\omega$ with $|\omega| = 3,$ there is some real $\lambda > 1$ such that $\omega,$ $\omega^2,$ and $\lambda \omega$ form an isosceles triangle in the complex plane. Find $\lambda$ if the isosceles triangle has two sides of equal length given by the distance between $\omega$ and $\omega^2$, and the distance between $\omega$ and $\lambda\omega$.

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