At most two elements give 1 to n

Question

Fix a positive integer $m$. Let $n$ ( $= n(m)$) be the largest positive integer for which there exists some subset $\{a_1,\ldots,a_m\} \subseteq \{1,2,\ldots,n\}$ of $m$ positive integers between $1$ and $n$ such that for every integer $1 \leq b \leq n$, there are $1 \leq i \leq j \leq m$ and $\lambda_1,\lambda_2 \in \{-1,0,1\}$ with $b=\lambda_1 a_i + \lambda_2 a_j$. I have got $\frac{m^2}{2}+O(m)\le n\le\frac{m^2}{1+\frac{2}{3\pi}}+O(m)$. Can we improve the bounds?