ÖMO 2009 Problem 1

If $x,y,K,m \in N$ , let us define: $a_m= \underset{k twos}{2^{2^{,,,{^{2}}}}}$ , $A_{km} (x)= \underset{k twos}{ 2^{2^{,,,^{x^{a_m}}}}}$ , $B_k(y)= \underset{m fours}{4^{4^{4^{,,,^{4^y}}}}}$ ,

Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$ , such that $A_{km} (x)=B_k(y)$