
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
In a plane, two vectors \(\overrightarrow{O A}\) and \(\overrightarrow{O B}\) satisfy \(|\overrightarrow{O A}|=a\) and \(|\overrightarrow{O B}|=b\), with \(a^2 + b^2 = 4\), and \(\overrightarrow{O A} \cdot \overrightarrow{O B} = 0\). If the vector \(\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}\) \((\lambda, \mu \in \mathbb{R})\) satisfies
\[
\left(\lambda-\frac{1}{2}\right)^{2} a^{2}+\left(\mu-\frac{1}{2}\right)^{2} b^{2}=1,
\]
then the maximum value of \(|\overrightarrow{O C}|\) is __________.

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