
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
(1) Given $f(x)=\begin{cases}\log_{2}x, & x > 0 \\ 2^{x}, & x \leq 0 \end{cases}$, find the value of $f(f(\frac{1}{8}))$.

(2) If the terminal sides of angles $α$ and $β$ are symmetric about the line $x + y = 0$, and $α = -\frac{π}{3}$, find the set of angles $β$.

(3) If $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{2}$, $\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}$, and $\overrightarrow{c} \perp \overrightarrow{a}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

(4) Given the line $y = a (0 < a < 1)$ and the function $f(x) = \sin ωx$, the first 12 intersection points' abscissas on the right side of the $y$-axis are $x_1$, $x_2$, $x_3$, ..., $x_{12}$, where $x_1 = \frac{π}{4}$, $x_2 = \frac{3π}{4}$, $x_3 = \frac{9π}{4}$. Find the sum of $x_1 + x_2 + x_3 + ... + x_{12}$.

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