
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
Given \( k \in \mathbb{N}, S_{k} = \left\{(a, b) \mid a, b = 1,2, \cdots, k \right\} \). For two elements \((a, b)\) and \((c, d)\) in \( S_{k} \), if
\[ a-c \equiv 0 \text { or } \pm 1, b-d \equiv 0 \text { or } \pm 1(\bmod k), \]
then \((a, b)\) and \((c, d)\) are called indistinguishable in \( S_{k} \). Otherwise, they are called distinguishable.

Consider the subset \( A \) of \( S_{k} \) that has the following property: every element in \( A \) is distinguishable from every other element in \( A \) within \( S_{k} \). The maximum number of elements in such a subset is denoted as \( r_{k} \).

1. Find \( r_{5} \) and explain the reason;
2. Find \( r_{7} \) and explain the reason;
3. What is \( r_{k} \) for general \( k \) (no need to provide a reason)?

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