
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 $a_1, a_2, \ldots, a_n$ be a permutation of $1, 2, \ldots, n$. The number of elements that are to the left of $a_i$ and smaller than $a_i$ is called the inversion number of $a_i$ ($i=1,2,\ldots,n$). For example, in the permutation $6, 4, 5, 3, 2, 1$, the inversion number of $5$ is $1$, and the inversion number of $3$ is $0$. Find the number of different permutations of the numbers $1, 2, 3, 4, 5, 6, 7, 8$ such that the inversion number of $8$ is $2$, the inversion number of $7$ is $3$, and the inversion number of $5$ is $3$. (Express the result as a number.)

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