
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 the function 
\[ f(x) = \begin{cases} 
\frac{1}{2} x + 1, & 0 < x < \frac{1}{2}, \\ 
2^{-4x} + 1, & \frac{1}{2} \leq x < 1, 
\end{cases} \]
find the solution set for \( f(x) > \frac{\sqrt{2}}{8} + 1 \).

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