
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 \( n \in \mathbf{N}^{*} \). Let \( d(n) \) denote the number of all positive divisors of \( n \) and \( \varphi(n) \) denote the number of integers among \( 1,2, \ldots, n \) 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 \( c \), find all positive integers \( n \) that satisfy the equation.

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