
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
Let \( A, B, C \) be distinct points on the circle \( O \) (where \( O \) is the origin of coordinates), and \(\angle AOB = \frac{2\pi}{3}\). If \(\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB}\) (\(\lambda, \mu \in \mathbf{R}\)), find \(\frac{\lambda}{\mu}\) when \(\omega = \sqrt{3} \lambda + \lambda + \mu\) reaches its maximum value.

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