
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 function $f(x)$ defined on $(0,+\infty)$, for $\forall x$, $y \gt 0$, it satisfies $f(x+y)=f(x)+f(y)-2$, $f(3)=4$, and for $\forall x_{1}$, $x_{2} \gt 0$, $\frac{f({x}_{1})-f({x}_{2})}{{x}_{1}-{x}_{2}}＞0$. Determine the solution set of the inequality $f(a^{2}-2a-3)≥\frac{8}{3}$.

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