
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 \( i_{1}, i_{2}, \cdots, i_{n} \) be a permutation of the set \(\{1,2, \cdots, n\}\). If there exist \(k < l\) such that \(i_{k} > i_{l}\), then the pair \(( i_{k}, i_{l} )\) is called an inversion. The number of all inversion pairs in a permutation is called the inversion number of that permutation. For instance, the permutation 1432 has inversions \(43, 42, 32\), so its inversion number is 3. Given \(n = 6\) and \(i_{3} = 4\), find the sum of the inversion numbers of all such permutations.

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