
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 that the monotonic function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x\_1 + x\_2) = f(x\_1) + f(x\_2)$ for any $x\_1$, $x\_2$. If positive real numbers $a$, $b$ satisfy $f(a) + f(2b - 1) = 0$, find the minimum value of $\frac{1}{a} + \frac{8}{b}$.

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