
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
Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer \(k\), denote by \(S_{k}\) the set of all visible lattice points \((x, y)\) such that \(x^{2}+y^{2}=k^{2}\). Let \(D\) denote the set of all positive divisors of \(2021 \cdot 2025\). Compute the sum
$$
\sum_{d \in D}\left|S_{d}\right|
$$

Here, a lattice point is a point \((x, y)\) on the plane where both \(x\) and \(y\) are integers, and \(|A|\) denotes the number of elements of the set \(A\).

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