
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 $a \in \mathbb{R}$. Consider the following functional equation:

$$
f(0)=0, \quad f(1)=1, \quad \text{if } x \leq y, \quad f\left(\frac{x+y}{2}\right)=(1-a)f(x)+a f(y)
$$

Find the values of $a$ for which the equation has at least one solution from $\mathbb{R}$ to $\mathbb{R}$. The same question applies when we require a continuous solution.

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