
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  $k \in \mathbb{N},$   $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$  Any two elements  $(a, b)$ ,  $(c, d)$   $\in S_k$  are called "undistinguishing" in  $S_k$  if  $a - c \equiv 0$  or  $\pm 1 \pmod{k}$  and  $b - d \equiv 0$  or  $\pm 1 \pmod{k}$ ; otherwise, we call them "distinguishing". For example,  $(1, 1)$  and  $(2, 5)$  are undistinguishing in  $S_5$ . Considering the subset  $A$  of  $S_k$  such that the elements of  $A$  are pairwise distinguishing. Let  $r_k$  be the maximum possible number of elements of  $A$ .

(i) Find  $r_5$ .
(ii) Find  $r_7$ .
(iii) Find  $r_k$  for  $k \in \mathbb{N}$ .

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