Given a finite sequence $S=(a_1,a_2,\dots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence

$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\dots ,\frac{a_{n-1}+a_n}{2}\right)$

of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m,2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0,$ and let $S=(1,x,x^2,\dots ,x^{60})$. If $A^{60}(S)=(1/2^{40})$, then what is $x$?