
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 ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left(a > b > 0\right)$ with its left vertex at $A$, left focus at $F$, and $P$ as a moving point on the ellipse. If the distance from the top vertex of the ellipse to the focus is $2$, and the eccentricity $e= \dfrac{1}{2}$, then the range of values for $\overrightarrow{AP} \cdot \overrightarrow{FP}$ is __________.

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