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Shelves are common in homes, warehouses, and commercial settings due to their storage efficiency. However, this efficiency comes at the cost of reduced visibility and accessibility. When looking from a side (lateral) view of a shelf, most objects will be fully occluded, resulting in a constrained lateral-access mechanical search problem. To address this problem, we introduce: (1) a novel bluction tool, which combines a thin pushing blade and suction cup gripper, (2) an improved LAX-RAY simulation pipeline and perception model that combines ray-casting with 2D Minkowski sums to efficiently generate target occupancy distributions, and (3) a novel SLAX-RAY search policy, which optimally reduces target object distribution support area using the bluction tool. Experimental data from 2000 simulated shelf trials and 18 trials with a physical Fetch robot equipped with the bluction tool suggest that using suction grasping actions improves the success rate over the highest performing push-only policy by 26% in simulation and 67% in physical environments.