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

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