
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 that the terminal side of angle $\alpha$ lies in the first quadrant, the domain of function $f(x)$ is $[0,1]$, and $f(0)=0$, $f(1)=1$, when $x \geqslant y$, there is $f(\frac{x+y}{2}) = f(x) \sin \alpha + (1 - \sin \alpha) f(y)$. Determine the set of values for $\alpha$ that makes the equation $f(\frac{1}{4}) = \frac{1}{4}$ hold true.

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