
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
Given \( n \in \mathbf{N}_{+} \), let \( d(n) \) represent the number of all positive divisors of \( n \), and \( \varphi(n) \) represent the number of integers from \( 1, 2, \ldots, n \) that are coprime with \( n \). Find all non-negative integers \( c \) such that there exists a positive integer \( n \) satisfying \( d(n) + \varphi(n) = n + c \), and for each such \( c \), determine all positive integers \( n \) that satisfy this equation.

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