
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
We say that an integer  $a$  is a quadratic, cubic, or quintic residue modulo  $n$  if there exists an integer  $x$  such that  $x^2\equiv a \pmod n$ ,  $x^3 \equiv a \pmod n$ , or  $x^5 \equiv a \pmod n$ , respectively. Further, an integer  $a$  is a primitive residue modulo  $n$  if it is exactly one of these three types of residues modulo  $n$ .

How many integers  $1 \le a \le 2015$  are primitive residues modulo  $2015$ ?

*Proposed by Michael Ren*

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