
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
Determine the smallest positive odd integer \( n \) such that the product \( 3^{\frac{1}{5}}3^{\frac{3}{5}}\cdots3^{\frac{2n+1}{5}} \) exceeds 5000. Note that the powers form an arithmetic sequence where the denominators of the exponents are all fives, and the numerators are the successive odd integers from 1 to \( 2n+1 \).
A) 5
B) 6
C) 7
D) 9

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