
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 positive integer \( n \), determine how many ordered quadruples of non-negative integers \(\left( k_{1}, k_{2}, k_{3}, k_{4} \right)\) there are such that after sawing off a segment of length \( k_{i} \) from each leg \( L_{i} \) (starting from the ground), for \( i = 1, 2, 3, 4 \), the table remains stable. Here, a table is considered stable if and only if it is possible to place one of the table's legs on the ground.

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