1. An in/out-of-gamut determination system comprising: an image forming apparatus; and an in/out-of-gamut determination device, wherein the in/out-of-gamut determination device includes: a display; a storage device including a hard disk drive that stores therein an in/out-of-gamut determination program; and a controller that includes a central processing unit, read only memory, and random access memory and that executes the in/out-of gamut determination program stored in the storage device to function as: a color gamut boundary line segment acquiring section that acquires a plurality of color gamut boundary line segments for defining an outermost shell of a color gamut of the image forming apparatus; a cross-section boundary line segment acquiring section that acquires a plurality of cross-section boundary line segments for defining a cross-section boundary as a boundary of a cross-section of the color gamut of the image forming apparatus; and an endpoint acquiring section that acquires endpoints of each of the plurality of cross-section boundary line segments, the color gamut boundary line segment acquiring section: prepares data of a chart to be printed that includes at least patches of colors of the outermost shell of the color gamut of the image forming apparatus; causes the image forming apparatus to print the chart based on the data; acquires Lab values of the patches of the colors of the outermost shell of the color gamut of the image forming apparatus; stores a correspondences between color values and the Lab values on a color-by-color basis with respect to the colors of the outermost shell of the color gamut of the image forming apparatus; acquires line segments each including two adjacent color values of the stored color values as endpoints thereof; and acquires line segments in a Lab color space corresponding to the acquired line segments in a RGB color space, the endpoint acquiring section: acquires an equation of a specific plane for acquiring the cross-section of the color gamut of the image forming apparatus; extracts line segments in parallel with the acquired specific plane among the acquired line segments in the Lab color spacer; and acquires intersection points between the acquired line segments in the Lab color space and the acquired specific plane to use the intersection points as the endpoints of each of the plurality of cross-section boundary line segments, the cross-section boundary line segment acquiring section: calls up a library of traveling salesman problem (TSP) solver; acquires the shortest possible route passing through each of the acquired endpoints exactly once using the library of the TSP solver; acquires line segments that make up the shortest possible route as provisional cross-section boundary line segments; chooses one of the acquired provisional cross-section boundary line segments as a target provisional cross-section boundary line segment; acquires a verification plane intersecting with the target provisional cross-section boundary line segment at a point other than the endpoints; acquires intersection points between the acquired line segments in the Lab color space and the acquired verification plane as endpoints of verification line segments of the line segments for verification of the target provisional cross-section boundary line segment; acquires the shortest possible route passing through each of the acquired endpoints of the verification line segments exactly once using the library of the TSP solver; acquires line segments that make up the shortest possible route as the verification line segments; determines whether or not the verification line segments and the target provisional cross-section boundary line segment intersect with one another; excludes the target provisional cross-section boundary line segment from the provisional cross-section boundary line segments and acquires the target provisional cross-section boundary line segment as a cross-section boundary line segment when determining that the acquired verification line segments and the target provisional cross-section boundary line segment intersect with one another; and excludes the target provisional cross-section boundary line segment from the provisional cross-section boundary line segments when determining that the acquired verification line segments and the target provisional cross-section boundary line segment do not intersect with one another, and the controller causes the display to display a cross-section boundary using the cross-section boundary line segments, and determines whether a specified Lab value is inside or outside the cross-section boundary.