Effective rate constants I have the following reversible bimolecular reactions

$$\begin{align}
\ce{A_0 +B &amp;&lt;=&gt;[$3k_+$][$k_-$] A_1} \\[5pt]
\ce{A_1 +B &amp;&lt;=&gt;[$2k_+$][$2k_-$] A_2} \\
\ce{A_2 +B &amp;&lt;=&gt;[$k_+$][$3k_-$] A_3} \\
\end{align}$$

that can be written as follows

$$\begin{align}
\ce{A_0 +3B &amp;&lt;=&gt;[$k_1$][$k_2$] A_3} \\[5pt]
\end{align}$$

The initial conditions are $[\ce{A_1}]_0 = [\ce{A_2}]_0 = [\ce{A_3}]_0 = 0$. I know rate constants $k_+$ and $k_-$ and would like to write rate constants $k_1$ and $k_2$ in terms of $k_+$ and $k_-$.

It would be great if you have any ideas. I am quite new to the topic. So far, I can only think of using some fitting algorithms to find $k_1$ and $k_2$.
Thanks!
Finding the equilibrium constant of your overall reaction is straightforward, as you've done in the comments, and we have $$K_\text{tot} = K_1K_2K_3 = \left(\frac{3k_+}{k_-}\right)\left(\frac{2k_+}{2k_-}\right)\left(\frac{k_+}{3k_-}\right) = \frac{k_+^3}{k_-^3}.$$ But this is insufficient to conclude that $k_1 = k_+^3$ or that $k_2 = k_-^3$. Indeed, let us consider what is stated when we have, as in your last reaction, rate  constants for the overall reaction. Such a scenario presumes an overall rate law of the form $$\frac{\text{d}\ce{[A_3]}}{\text{d}t} = k_1\ce{[A_0][B]^3} - k_2\ce{[A_3]},$$ and there is no a priori reason to suggest that this is likely. (Often this is possible in certain limiting regimes, but not in general.) To investigate $k_1$ and $k_2$, we will have to solve our system fully. We can write out the rate equations for our system as 
\begin{align*}
\frac{\text{d}\ce{[A_1]}}{\text{d}t} &amp;= 3k_+\ce{[A_0][B]}-k_-\ce{[A_1]}-2k_+\ce{[A_1][B]}+2k_-\ce{[A_2]}\\
\frac{\text{d}\ce{[A_2]}}{\text{d}t} &amp;= 2k_+\ce{[A_1][B]}-2k_-\ce{[A_2]}-k_+\ce{[A_2][B]}+3k_-\ce{[A_3]}\\
\frac{\text{d}\ce{[A_3]}}{\text{d}t} &amp;= k_+\ce{[A_2][B]}-3k_-\ce{[A_3]},
\end{align*}
a coupled system of differential equations, which can be solved using matrix methods---diagonalize the coupling matrix, find the eigenvectors and eigenvalues, and solve---but these calculations are often quite messy, so I will leave them to you if you're interested.

Once $\ce{[A_2]}$ and $\ce{[A_3]}$ are known, we can plug them into the last differential equation and compare it with the presumed form of the overall rate law to determine $k_1$ and $k_2$.