  We prove that any cyclic Nakayama algebra which is a higher Auslander algebra
can be uniquely constructed from Nakayama algebras of smaller ranks by
reversing the syzygy filtration process. This creates chains of higher
Auslander algebras upto $\boldsymbol\varepsilon$-equivalences. Therefore, the
classification of all cyclic Nakayama algebras which are higher Auslander
algebras reduces to the classification of linear ones. We give two applications
of this: for any integer $k$ where $2\leq k\leq 2n-2$, there is a Nakayama
algebra of rank $n$ which is a higher Auslander algebra of global dimension $k$
and the possible values of the global dimensions of cyclic Nakayama algebras
which are higher Auslander algebras form the sets
$\left\{2,\ldots,2n-2\right\}\setminus\left\{n-1\right\}$ if $n$ is even and
$\left\{2,\ldots,2n-2\right\}\setminus\left\{ 2,n-1\right\}$ if $n$ is odd.
