
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 hyperbola $\frac {x^{2}}{4} - \frac {y^{2}}{b^{2}} = 1\ (b > 0)$ with an eccentricity of $\frac { \sqrt {5}}{2}$, where F<sub>1</sub> and F<sub>2</sub> are the two foci of the hyperbola, A is the left vertex, B is the point (0, b), and point P lies on segment AB, find the minimum value of $\overrightarrow {PF_{1}} \cdot \overrightarrow {PF_{2}}$.

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