
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 vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$ satisfy $|\overrightarrow{OA}|=|\overrightarrow{OB}|=1$, $\overrightarrow{OA} \cdot \overrightarrow{OB}=0$, $\overrightarrow{OC}=\lambda \overrightarrow{OA}+\mu \overrightarrow{OB} (\lambda,\mu \in \mathbb{R})$. If $M$ is the midpoint of $AB$, and $|\overrightarrow{MC}|=1$, determine the equation of the trajectory of the point $(\lambda,\mu)$.

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