请解答一下图片中的题目，要求步骤详细，通俗易懂---强化训练13
如图，一条宽为 $\sqrt{2}$ ($AB=\sqrt{2}$) 的长方形纸带，$E, F$ 分别是边 $AD, BC$ 上的点，$\angle DEF=\alpha$，将纸带沿 $EF$ 折叠成图 1，再沿 $GF$ 折叠成图 2。
(1) 如图 3, 当点 $F$ 与点 $C$ 重合, 且 $\alpha=45^\circ$ 时, 则 $S_{\triangle EFG}=$ ________ °。
(2) 如图 2, 若 $\angle GFE=\frac{1}{3}\angle GFN$ 时, 求 $\alpha$ 的值。
(3) 作 $GP$ 平分 $\angle MGF$, 交直线 $EF$ 于点 $P$。
① 如图 4, 若 $\alpha<60^\circ$, 试猜想 $\angle GEP$ 与 $\angle GPE$ 的数量关系, 并说明理由。
② 若 $60^\circ<\alpha<90^\circ$, 直接写出 $\angle GEP$ 与 $\angle GPE$ 的数量关系。
图 1
图 2
图 3
图 4
Chart/Diagram Description:
The image contains four geometric diagrams labeled as 图 1, 图 2, 图 3, and 图 4, illustrating the folding process of a rectangular paper strip.
Diagram 1:
Type: Geometric figure.
Shows a rectangle (labeled ABGD') which seems to be a part of the original strip after a first fold along EF. A point C' is shown, which is likely the original position of C after folding. Points E, F are on AD and BC respectively. Points G, D' and C' are likely results of the first fold. AD and BC are parallel. Lines EF, EG, FG, GD', FC' are shown. Dashed lines indicate original positions or constructions (D-D', C-C'). Angle DEF is labeled as α.
Diagram 2:
Type: Geometric figure.
Shows the result after a second fold along GF. The folded part MEG seems to overlap with FNG. Point M and N are introduced. Lines EF, FG, EG, MG, NG are shown. Dashed lines indicate original positions (D-D', C-C'). Angle GFE is labeled. Angle GFN is indicated.
Diagram 3:
Type: Geometric figure.
Shows a state where point F coincides with point C. The original rectangular strip is shown with points A, B, E, G, F(C), D. Line EF(C) is the fold line. EG is shown. A right angle is indicated at B and C. Dashed lines from E to D and from F(C) to D are shown.
Diagram 4:
Type: Geometric figure.
Shows a diagram related to part (3). It is similar to Diagram 2 but includes point P on line EF. A line GP is shown as bisecting angle MGF. Points A, B, E, G, F, M, N, P, D', C' are labeled. Dashed lines D-D' and C-C' are shown. Lines EF and GP intersect at P.