
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 a positive integer \( n \) (where \( n \geq 2 \)), consider an \( n \)-layer inverted triangle letter table that meets the following conditions:
1. The \( i \)-th row consists of \( n+1-i \) letters, each being one of \( A, J, Z \).
2. Any three adjacent letters forming a smaller inverted triangle must either all be different letters or all the same letter.

If a letter table's three vertices are either all different letters or all the same letter, it is called "good". Find all positive integers \( n \) such that every letter table satisfying the conditions is necessarily "good".

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