
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 a monotonic function $f(x)$ defined on $\mathbb{R}$, satisfying $f(x_1+x_2)=f(x_1)+f(x_2)$ for any real numbers $x_1, x_2$. If positive real numbers $a, b$ meet the condition $f(a)+f(2b-1)=0$, then the minimum value of $\frac{1}{a} + \frac{8}{b}$ is _______.

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