
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  $\sigma(n)$  be the number of positive divisors of  $n$ , and let  $\operatorname{rad} n$  be the product of the distinct prime divisors of  $n$ . By convention,  $\operatorname{rad} 1 = 1$ .  Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural*

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