Part $2$

$a_{n+1}=\frac{a_n+g_n}{2}$

$g_{n+1}=\sqrt[\phantom{2}]{a_n{g}_n}$

the sequence $(g_n)$ with $g_0=y.$

For example, if $x=y=1$ then $a_n=1$ and $g_n=1$ for all $n,$ so agm$\_$sequences $(1,1,3)$ must return

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Kernel...

$[1,1,1],[1,1,1]$

and if $x=2,y=8$ then $a_0=2,g_0=8,a_1=5,g_1=4$ so agm$\_$sequences $(2,8,2)$ must return

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$[(2,5],[8,4]]$