
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  $p=2017$  be a prime. Suppose that the number of ways to place  $p$  indistinguishable red marbles,  $p$  indistinguishable green marbles, and  $p$  indistinguishable blue marbles around a circle such that no red marble is next to a green marble and no blue marble is next to a blue marble is  $N$ . (Rotations and reflections of the same configuration are considered distinct.) Given that  $N=p^m\cdot n$ , where  $m$  is a nonnegative integer and  $n$  is not divisible by  $p$ , and  $r$  is the remainder of  $n$  when divided by  $p$ , compute  $pm+r$ .

*Proposed by Yannick Yao*

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