
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  $\mathbb{N}$  be the set of positive integers. Let  $n\in \mathbb{N}$  and let  $d(n)$  be the number of divisors of  $n$ . Let  $\varphi(n)$  be the Euler-totient function (the number of co-prime positive integers with  $n$ , smaller than  $n$ ).
 
Find all non-negative integers  $c$  such that there exists  $n\in\mathbb{N}$  such that \[ d(n) + \varphi(n) = n+c , \] and for such  $c$  find all values of  $n$  satisfying the above relationship.

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