
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
Suppose a positive integer \(n \geq 3\). If there are \(n\) grid points \(P_1, P_2, \cdots, P_n\) on a plane such that:
- When \(\left|P_i P_j\right|\) is a rational number, there exists a point \(P_k\) such that both \(\left|P_i P_k\right|\) and \(\left|P_j P_k\right|\) are irrational numbers;
- When \(\left|P_i P_j\right|\) is an irrational number, there exists a point \(P_k\) such that both \(\left|P_i P_k\right|\) and \(\left|P_j P_k\right|\) are rational numbers,

then \(n\) is called a "good number".  
(1) Find the smallest good number.  
(2) Determine if 2005 is a good number and explain your reasoning.

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