
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
Given a circle with center $O$ and radius $1$, there are three distinct points $A$, $B$, and $C$ on the circle, where $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. There exist real numbers $\lambda$ and $\mu$ satisfying $\overrightarrow{OC} + \lambda \overrightarrow{OA} + \mu \overrightarrow{OB} = \overrightarrow{0}$. Derive the relationship between the real numbers $\lambda$ and $\mu$.

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