
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 \( f:(0,1) \rightarrow (0,1) \) be a differentiable function with a continuous derivative such that for every positive integer \( n \) and odd positive integer \( a < 2^n \), there exists an odd positive integer \( b < 2^n \) such that \( f\left(\frac{a}{2^n}\right) = \frac{b}{2^n} \). Determine the set of possible values of \( f'\left(\frac{1}{2}\right) \).

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