HMMT 2012 T9

For any positive integer $n$, let $N=\varphi(1)+\varphi(2)+\ldots+\varphi(n)$. Show that there exists a sequence

$$a_{1}, a_{2}, \ldots, a_{N}$$

containing exactly $\varphi(k)$ instances of $k$ for all positive integers $k \leq n$ such that

$$\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{N} a_{1}}=1 .$$