<image>[Math Problem] (Total Score: 13 points)
A viewing platform P is located on the top of a mountain with a vertical height PO of 2 km from the mountain top to the horizontal plane M at the foot of the mountain. At the foot of the mountain, there is a straight road AB lying on the horizontal line. The mountain slope surface can be approximately considered as a plane PAB, and △PAB is an isosceles triangle. The dihedral angle between the mountain slope surface and the horizontal plane M at the foot of the mountain is α (0° < α < 90°), and sinα = . Now, a winding mountain road is to be built starting from a certain point C0 on the horizontal road AB at the foot of the mountain. The first segment, second segment, third segment, ..., (n-1)th segment of this road are sequentially labeled as C0C1, C1C2, C2C3, ..., C(n-1)Cn (as shown in the figure), and each segment C0C1, C1C2, C2C3, ..., C(n-1)Cn forms an angle β with AB, where 0 < β < 90°, and sinβ = . The questions are:

(1) How many meters of winding mountain road must be built to increase the vertical height by 100 meters? If the winding mountain road is built up to the halfway point of the mountain (half the height of the mountain), and a cable car station is built at the center point Q of the halfway point, with the cable car PQ being built along the mountain (parallel to the mountain slope, height difference from the slope is negligible), what are the lengths of the winding mountain road and the cable car, respectively?
(2) If the cost to build x km of the winding mountain road is a ten thousand yuan. The cost to build the cable car is 2a ten thousand yuan per km. To minimize the total cost, to what height should the winding mountain road be built before constructing the cable car to the viewing platform?