18. (This question is worth 12 points) As shown in Figure 4, the right triangular prism $\mathrm { ABC } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ has an equilateral triangle base with side length 2. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { BC }$ and $\mathrm { CC } _ { 1 }$ respectively. (I) Prove that: plane $\mathrm { AEF } \perp$ plane $\mathrm { B } _ { 1 } \mathrm { BCC } _ { 1 }$ (II) If the angle between line $A _ { 1 } C$ and plane $A _ { 1 } A B B _ { 1 }$ is $45 ^ { \circ }$, find the volume of the triangular pyramid F-AEC. [Figure]
18. (This question is worth 12 points)\\
As shown in Figure 4, the right triangular prism $\mathrm { ABC } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ has an equilateral triangle base with side length 2. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { BC }$ and $\mathrm { CC } _ { 1 }$ respectively.\\
(I) Prove that: plane $\mathrm { AEF } \perp$ plane $\mathrm { B } _ { 1 } \mathrm { BCC } _ { 1 }$\\
(II) If the angle between line $A _ { 1 } C$ and plane $A _ { 1 } A B B _ { 1 }$ is $45 ^ { \circ }$, find the volume of the triangular pyramid F-AEC.\\
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