  The purpose of this paper has twofold. The first is to establish a second
main theorem for meromorphic functions on annuli and meromorphic function
targets (may not be small functions) with truncated counting functions
(truncation level 1) and with a detailed estimate for the error term. The
second is to show that if the polynomial $$P_S(w)=(w-a_1)\cdots (w-a_q)$$ is a
uniqueness polynomial for admissible meromorphic functions on an annulus
$\mathbb A(R_0)$ such that $P'_S(w)$ has exactly $k$ distinct zeros and
$q>\frac{(5k+7)\ell}{2\ell-175}$, then the set $S=\{a_1,\ldots,a_q\}$ is a
finite range set with truncation level $\ell$ for admissible meromorphic
functions on $\mathbb A(R_0)$. This result extends the previous result on the
finite range set (with truncation level $\ell=\infty$) for holomorphic
functions on $\mathbb C$ of H. Fujimoto.
