One. (20 points) Given the equations about $x$: $2 x^{2}-4 n x$ $-2 n=1$ and $x^{2}-(3 n-1) x+2 n^{2}-3 n=2$. Does there exist a value of $n$ such that the sum of the squares of the two real roots of the first equation equals an integer root of the second equation? If it exists, find such $n$ values; if not, explain the reason.