
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 $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions 
(a) $0\le A<B<C\le99$, 
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\le b<a<c<p$, 
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and 
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$.

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