
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
For all rational \( x \) satisfying \( 0 \leq x < 1 \), the function \( f \) is defined by
\[ 
f(x) = 
\begin{cases} 
\frac{f(2x)}{4}, & \text{for } 0 \leq x < \frac{1}{2}, \\
\frac{3}{4} + \frac{f(2x - 1)}{4}, & \text{for } \frac{1}{2} \leq x < 1.
\end{cases}
\]
Given that \( x = 0.b_1 b_2 b_3 \ldots \) is the binary representation of \( x \), find \( f(x) \).

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