
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 two vectors $\overrightarrow {PA}$ and $\overrightarrow {PB}$ on a plane that satisfy $\overrightarrow {PA}^2 + \overrightarrow {PB}^2 = 4$ and $\overrightarrow {PA} \cdot \overrightarrow {PB} = 0$, if $\overrightarrow {PC} = \frac{1}{3}\overrightarrow {PA} + \frac{2}{3}\overrightarrow {PB}$, find the maximum value of $|\overrightarrow {PC}|$.

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