
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 $\frac{x^{2}}{4}+y^{2}=1$ with foci $F\_1$ and $F\_2$, let $M$ be any point on the major axis $A\_1A\_2$. Draw a line perpendicular to $A\_1A\_2$ through $M$, intersecting the ellipse at point $P$. Determine the probability that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}} < 0$ for point $M$.

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