
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 \( M = \{1, 2, \cdots, 2005\} \), and \( A \) be a subset of \( M \). If for any \( a_i, a_j \in A \) with \( a_i \neq a_j \), an isosceles triangle can be uniquely determined with \( a_i \) and \( a_j \) as side lengths, find the maximum value of \( |A| \).

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