
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 \( g(n) \) denote the sum of all the positive divisors of the integer \( n \) including twice the greatest prime divisor of \( n \). How many integers \( j \) exist such that \( 1 \leq j \leq 5000 \) and \( g(j) = j + 2\sqrt{j} + 1 \)?

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