
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
Given the function \( f(x) = A \sin (\omega x + \varphi) \) where \( A \) and \( \varphi \) are real numbers, find the number of elements in the set of positive integer values of \( \omega \) that make the smallest positive period \( T \) of \( f(x) \) fall within the interval \( \left(\frac{1}{100}, \frac{1}{50}\right) \).

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