
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  $p = 2017,$  a prime number. Let  $N$  be the number of ordered triples  $(a,b,c)$  of integers such that  $1 \le a,b \le p(p-1)$  and  $a^b-b^a=p \cdot c$ . Find the remainder when  $N$  is divided by  $1000000.$ *Proposed by Evan Chen and Ashwin Sah*

*Remark:* The problem was initially proposed for  $p = 3,$  and  $1 \le a, b \le 30.$ 

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