Finding an Integral vector in an Unknown Polyhedral Cone

We present an algorithm to find an integral vector in the polyhedral cone $\mathrm{\Gamma }=\{X\mathrm{\mid }\mathbf{\text{A}}X\le \mathbf{\text{0}}\}$, without assuming the explicit knowledge of $\mathbf{\text{A}}$. About the polyhedral cone, $\mathrm{\Gamma }$, it is only given that, (i) the elements of $\mathbf{\text{A}}$ are in $\{-d,-d+1,\dot{\mathrm{.}}\mathrm{.},0,\dot{\mathrm{.}}\mathrm{.},d-1,d\}$, $d\in \mathbb{N}$, and, (ii) $Y=[y(1),y(2),\dot{\mathrm{.}}\mathrm{.},y(n)]$ is a non-zero integral solution to $\mathrm{\Gamma }$. The proposed algorithm finds a non-zero integral vector in $\mathrm{\Gamma }$ such that its maximum element is less than $(2d{)}^{2^{n-1}-1}\mathrm{/}{2}^{n-1}$.