
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
Legs \( L_{1}, L_{2}, L_{3}, L_{4} \) of a square table each have length \( n \), where \( n \) is a positive integer. For how many ordered 4-tuples \( \left(k_{1}, k_{2}, k_{3}, k_{4}\right) \) of nonnegative integers can we cut a piece of length \( k_{i} \) from the end of leg \( L_{i} \) and still have a stable table?
(The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)

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