
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 $\overrightarrow {OA}$ and $\overrightarrow {OB}$ satisfy $|\overrightarrow {OA}|=|\overrightarrow {OB}|=1$, $\overrightarrow {OA} \perp \overrightarrow {OB}$, $\overrightarrow {OC} = \lambda \overrightarrow {OA} + \mu \overrightarrow {OB}$ ($\lambda, \mu \in \mathbb{R}$). If $M$ is the midpoint of segment $AB$, and $|\overrightarrow {MC}|=1$, calculate the maximum value of $\lambda + \mu$.

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