
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 \( n \), suppose real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) and positive real numbers \( b_{1}, b_{2}, \cdots, b_{n} \) satisfy
\[ 
\sum_{i=1}^{n} a_{i} \geq 8 \quad \text{and} \quad \sum_{i=1}^{n} b_{i} \leq 4.
\]
Find the minimum value of \( \prod_{i=1}^{n} \mathrm{e}^{\frac{(\max(0, a_{i}))^{2}}{b_{i}}} \).

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