
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
a) In the cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), a common perpendicular \( MN \) is drawn to the lines \( A_{1} B \) and \( B_{1} C \) (point \( M \) lies on the line \( A_{1} B \) ). Find the ratio \( A_{1} M : M B \).

b) Given a cube \( ABCD A_{1} B_{1} C_{1} D_{1} \). Points \( M \) and \( N \) are taken on the segments \( A A_{1} \) and \( B C_{1} \) respectively, such that the lines \( MN \) and \( B_{1} D \) intersect. Find the difference between the ratios \( B C_{1} : B N \) and \( A M : A A_{1} \).

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