
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 each positive integer \( n \), define the set
\[ P_{n}=\{n^{k} \mid k=0,1, \cdots\}. \]

For positive integers \( a, b, c \), if there exists a positive integer \( m \) such that the numbers \( a-1, a b-12, a b c-2015 \) (not necessarily distinct) are all elements of the set \( P_{m} \), then the positive integer tuple \( (a, b, c) \) is called "lucky". Find the total number of lucky tuples.

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