
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 cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with side length 101 is to be divided into two parts with the following requirements: (1) The dividing plane intersects each edge $AA_{1}, BB_{1}, CC_{1}, DD_{1}$ at points $P, Q, R, S$ (which may coincide with the vertices); (2) The lengths of segments $AP, BQ, CR, DS$ are all non-negative integers, and each set of values for segments $AP, BQ, CR, DS$ corresponds to a unique way of dividing the cube. Determine the number of different ways to divide the cube. Answer with a number.

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