
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
Given a cube \(A B C D A_{1} B_{1} C_{1} D_{1}\), points \(M\) and \(N\) are taken on the segments \(A A_{1}\) and \(B C_{1}\) respectively, such that the line \(M N\) intersects the line \(B_{1} D\). Find

$$
\frac{\left|B C_{1}\right|}{|B N|}-\frac{|A M|}{\left|A A_{1}\right|}
$$

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