
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 that $|$$\overrightarrow {a}$$| = 2|$ $\overrightarrow {b}$$|, $|$ $\overrightarrow {b}$$| \neq 0$, and the quadratic equation $x^2 + |$ $\overrightarrow {a}$$|x - $$ $\overrightarrow {a}$ \cdot$ $\overrightarrow {b}$ = 0 has two equal real roots, find the angle between the vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.

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