A solid cube with a side length of 10 inches has a cylindrical hole drilled through its center, with the cylinder having a radius of 2 inches and passing completely through the cube along one of its axes. Additionally, a conical section is removed from one of the faces of the cube, with the cone having a base radius of 2 inches and a height of 5 inches. What is the total remaining volume of the solid? Express your answer in cubic inches in terms of $\pi$ and round to the nearest whole number. Let's think step by step and output the final answer within \boxed{}.
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