
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 the set \( A = \{ z_1, z_2, \cdots, z_2024 \} \), where \( z_k \in \mathbb{C} \) and \( z_k \neq 0 \) for \( k = 1, 2, \cdots, 2024 \). If for any positive integers \( 1 \leq k, m \leq 2024 \), we have both
\[ \frac{z_m}{z_k} \text{ and } z_m z_k \in A, \]
determine the number of elements \( |B| \) in the set 
\[ B = \{ (z_x, z_y) \mid \operatorname{Re}(z_x + z_y) + \operatorname{Im}(z_x + z_y) > 0, 1 \leq x \leq 2024, 1 \leq y \leq 2024, x, y \in \mathbb{N} \}. \]

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