
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
Let \( L_{1}, L_{2}, L_{3}, L_{4} \) be the four legs of a square table, and their heights are all positive integers \( n \). How many ordered quadruples of non-negative integers \( (k_{1}, k_{2}, k_{3}, k_{4}) \) are there such that if each leg \( L_{i} \) is cut by a segment of length \( k_{i} \) (starting from the ground), where \( i = 1, 2, 3, 4 \), the table remains stable? Here, the table is considered stable if and only if all four legs can simultaneously touch the ground.

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