
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 square table has four legs $L_{1}, L_{2}, L_{3}, L_{4}$ each with a length of $n$, where $n$ is a positive integer. How many ordered quadruples $(k_{1}, k_{2}, k_{3}, k_{4})$ exist, such that if the legs $L_{i}$ are shortened by $k_{i}$ units $(i = 1, 2, 3, 4)$, the table remains stable? Here, $k_{1}, k_{2}, k_{3}, k_{4}$ are non-negative integers, and a stable table means that all four legs must touch the ground, allowing for a shortening length of zero.

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