A symmetric series

A recurrence relation for $m,n \ge 0$ is defined as follows:

• $A_{0n} =A_{n0} = n$
• $A_{mn}=(A_{m-1,n} + A_{m,n-1})^{\frac{8}{9}}$ for $m>1$ and $n>1$

Evaluate:

$\large \lim_{n \to \infty} A_{nn}$