
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 each integer \( n \geq 2 \), let \( A(n) \) be the area of the region in the coordinate plane defined by the inequalities \( 1 \leq x \leq n \) and \( 0 \leq y \leq x \left\lfloor \log_2{x} \right\rfloor \), where \( \left\lfloor \log_2{x} \right\rfloor \) is the greatest integer not exceeding \( \log_2{x} \). Find the number of values of \( n \) with \( 2 \leq n \leq 100 \) for which \( A(n) \) is an integer.

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