
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 \( ABCD A_{1} B_{1} C_{1} D_{1} \) with edge length \( a \). The bases of the cylinder are inscribed in the faces \( ABCD \) and \( A_{1}B_{1}C_{1}D_{1} \). Let \( M \) be a point on edge \( AB \) such that \( |AM| = a / 3 \), and \( N \) a point on edge \( B_{1}C_{1} \) such that \( |NC_{1}| = a / 4 \). A plane passes through points \( C_{1} \) and \( M \), tangent to the base of the cylinder inscribed in \( ABCD \), and another plane passes through points \( A \) and \( N \), tangent to the base of the cylinder inscribed in \( A_{1}B_{1}C_{1}D_{1} \). 

Find the volume of the part of the cylinder enclosed between the planes.

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