
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
a) Find the number of natural divisors of \(N=1 \underbrace{00 \ldots 0}_{40}\) that are neither perfect squares (i.e., squares of natural numbers) nor perfect cubes.

b) Find the number of natural divisors of \(N=1 \underbrace{00 \ldots 0}_{40}\) that cannot be represented in the form \(m^n\), where \(m\) and \(n\) are natural numbers and \(n > 1\).

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