
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]


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For any two different real numbers \( x \) and \( y \), we define \( D(x, y) \) to be the unique integer \( d \) satisfying \( 2^{d} \leq |x-y| < 2^{d+1} \). Given a set of real numbers \(\mathcal{F}\), and an element \( x \in \mathcal{F} \), we say that the scales of \( x \) in \(\mathcal{F}\) are the values of \( D(x, y) \) for \( y \in \mathcal{F} \) with \( x \neq y \).

Let \( k \) be a given positive integer. Suppose that each member \( x \) of \(\mathcal{F}\) has at most \( k \) different scales in \(\mathcal{F}\). What is the maximum possible size of \(\mathcal{F}\)?

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