
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
For  $R>1$  let  $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$ . Compute   $$ \lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}. $$  *Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro*

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