Question 125
Find all positive integers $n$ for which there exist positive integers $x_{1}, x_{2}, \ldots, x_{n}$ such that
$$
\frac{1}{x_{1}^{2}}+\frac{2}{x_{2}^{2}}+\frac{4}{x_{3}^{2}}+\cdots+\frac{2^{n-1}}{x_{n}^{2}}=1
$$
$$
\frac{1}{x_{1}^{2}}+\frac{2}{x_{2}^{2}}+\frac{4}{x_{3}^{2}}+\cdots+\frac{2^{n-1}}{x_{n}^{2}}=1
$$