
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
In \(\triangle ABC\), the incircle centered at \(I\) touches sides \(AB\) and \(BC\) at \(X\) and \(Y\), respectively. Additionally, the area of quadrilateral \(BXIY\) is \(\frac{2}{5}\) of the area of \(ABC\). Let \(p\) be the smallest possible perimeter of a \(\triangle ABC\) that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter \(p\).

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