
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 $F\_1$ and $F\_2$ are two foci of the ellipse $\frac{x^2}{4} + y^2 = 1$, $A$ and $B$ are the left and upper vertices of the ellipse respectively, and point $P$ is on the line 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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