
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
On a plane, two vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ satisfy $|\overrightarrow{OA}| = a$ and $|\overrightarrow{OB}| = b$, with $a^2 + b^2 = 4$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. Given the vector $\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB}$ ($\lambda, \mu \in \mathbf{R}$), and the condition $\left(\lambda - \frac{1}{2}\right)^2 a^2 + \left(\mu - \frac{1}{2}\right)^2 b^2 = 1$, determine the maximum value of $|\overrightarrow{OC}|$.

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