
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
For any positive integer  $n$ , we have the set  $P_n = \{ n^k \mid k=0,1,2, \ldots \}$ . For positive integers  $a,b,c$ , we define the group of  $(a,b,c)$  as lucky if there is a positive integer  $m$  such that  $a-1$ ,  $ab-12$ ,  $abc-2015$  (the three numbers need not be different from each other) belong to the set  $P_m$ . Find the number of lucky groups.

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