
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\\), ..., \\(a_n\\) be a permutation of the numbers \\(1\\), \\(2\\), ..., \\(n\\). Define the "order number" of \\(a_i\\) (for \\(i=1,2,…,n\\)) as the count of numbers to the left of \\(a_i\\) that are smaller than \\(a_i\\). For example, in the permutation \\(6\\), \\(4\\), \\(5\\), \\(3\\), \\(2\\), \\(1\\), the order number of \\(5\\) is \\(1\\), and the order number of \\(3\\) is \\(0\\). In all permutations formed by the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), \\(7\\), \\(8\\), find the number of distinct permutations where the order number of \\(8\\) is \\(2\\), the order number of \\(7\\) is \\(3\\), and the order number of \\(5\\) is \\(3\\). (Express the result as a number.)

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