
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
α is a fixed real number. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) (where \( \mathbb{R} \) is the reals) which are continuous, have a continuous derivative, and satisfy

\[ \int_y^b f(\alpha(x)) \, dx = \left( \int_y^b f(x) \, dx \right)^\alpha \]

for all \( y \) and some \( b \).

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