
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). A rectangle \( MNLK \) is inscribed in square \( ABCD \) such that one of its vertices is point \( M \) and the other three vertices are located on different sides of the square base. Rectangle \( M_1 N_1 L_1 K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane \( A_1 B_1 C_1 D_1 \) of the top face. The plane of quadrilateral \( MK_1 L_1 N \) forms an angle \( \alpha \) with the plane of the cube's base, where \( \cos \alpha=\sqrt{\frac{2}{11}} \). Find the ratio \( AM : MB \).

Present the answer in LaTex format: \boxed{Your answer}
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