
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 \( A = \{1, 2, \cdots, 2016\} \). For any 1008-element subset \( X \) of \( A \), if there exist \( x \) and \( y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good set". Find the largest positive integer \( a \) (where \( a \in A \)) such that any 1008-element subset containing \( a \) is a good set.

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