
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 \( p \in \mathbb{N} \) and \( n = 2^p \). Consider the set \( E = \{1, 2, \cdots, n\} \) with subsets \( A \) that have the following property: if \( x \in A \), then \( 2x \in A \). Find the maximum number of elements in such a subset \( A \).

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