
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 the integers $n \geq 3$, $a_{1}, a_{2}, \cdots, a_{n}$ be any $n$ different real numbers whose sum is positive. Let $b_{1}, b_{2}, \cdots, b_{n}$ be a permutation of these $n$ numbers. If for $k=1,2, \cdots, n$, $\sum_{i=1}^{k} b_{i} > 0$ holds, then $b_{1}, b_{2}, \cdots, b_{n}$ is called a "good permutation." Find the minimum number of "good permutations."

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