Putnam 1990 B4

Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence $$g_1, g_2, g_3, \dots, g_{2n}$$ such that

(1) every element of $G$ occurs exactly twice, and

(2) $g_{i+1}$ equals $g_i a$ or $g_i b$ for $i = 1, 2, \dots, 2n$. (Interpret $g_{2n+1}$ as $g_1$.)