
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 \( a(r) \) be the number of lattice points inside the circle centered at the origin with radius \( r \). Let \( k = 1 + e^{-1} + e^{-4} + \ldots + \exp(-n^2) + \ldots \). Express \( \int_U a(\sqrt{x^2 + y^2}) \exp\left(-(x^2 + y^2)\right) \, dx \, dy \) as a polynomial in \( k \), where \( U \) represents the entire plane.

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