
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 the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, where $\overrightarrow {a}$⋅($\overrightarrow {a}$+$\overrightarrow {b}$)=3 and $\overrightarrow {a}$=($\frac {1}{2}$, $\frac {\sqrt {3}}{2}$) and $|\overrightarrow {b}|$=$2 \sqrt {5}$, find the magnitude of $|\overrightarrow {a}$+$\overrightarrow {b}|$.

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