
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
In the set of rational numbers, define: \( f(0) = 0 \), \( f(1) = 1 \),
$$
f(x)=\left\{
\begin{array}{l}
\frac{f(2x)}{4}, \quad 0 < x < \frac{1}{2}; \\
\frac{3}{4} + \frac{f(2x-1)}{4}, \quad \frac{1}{2} \leq x < 1.
\end{array}
\right.
$$

If \( x \) is represented in binary as \( x = (0.b_1 b_2 b_3 \cdots)_2 \), find the binary representation of \( f(x) \).

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