
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
If we let $g(n)$ denote the sum of the cubes of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 500$ and $g(i) = 1 + \sqrt{i^3} + i^3$?

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