
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
For any two different real numbers \( x \) and \( y \), define \( D(x, y) \) as the unique integer \( d \) that satisfies
\[ 
2^{d} \leq |x-y| < 2^{d+1}.
\]
Given a set of real numbers \( \mathscr{F} \), for \( x \in \mathscr{F} \), the "scale" of \( x \) in \( \mathscr{F} \) is defined as the value of \( D(x, y) \) for some \( y \in \mathscr{F} \) where \( x \neq y \).

Let \( k \) be a given positive integer. Suppose each \( x \) in \( \mathscr{F} \) has at most \( k \) different scales in \( \mathscr{F} \) (these scales may vary depending on \( x \)). Find the maximum number of elements in the set \( \mathscr{F} \).

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