
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
Let \( P \) be a convex polygon. Let \( Q \) be the interior of \( P \) and \( S = P \cup Q \). Let \( p \) be the perimeter of \( P \) and \( A \) its area. Given any point \((x, y)\) let \( d(x, y) \) be the distance from \((x, y)\) to the nearest point of \( S \). Find constants \(\alpha, \beta, \gamma\) such that
\[
\int_{U} e^{-d(x, y)} \, dx \, dy = \alpha + \beta p + \gamma A,
\]
where \( U \) is the whole plane.

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