
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 \( p \) be a prime number, \( n \) be a positive integer, and \( n = n_0 + n_1 p + n_2 p^2 + \cdots + n_i p^t \), where \( n_i \in \mathbb{N}^*, 0 \leq n_i \leq p-1, i=0,1,2, \cdots, t \). Let \( S_n \) denote the set of ordered triples \( (a, b, c) \) that satisfy the following conditions:
1. \( a, b, c \) are non-negative integers;
2. \( a + b + c = n \);
3. \( \frac{n!}{a!b!c!} \) is not divisible by \( p \).

How many ordered triples \( (a, b, c) \) are in the set \( S_n \)?

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