
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, \( n \in \mathbb{Z}_{+} \), and

\[ n = n_{0} + n_{1} p + \cdots + n_{t} p^{t}, \]

where \( n_{i} \in \mathbb{N} \), \( 0 \leq n_{i} \leq p-1 \) for \( i = 0, 1, \ldots, t \).

Let \( S_{n} \) be the set of ordered triples \((a, b, c)\) that satisfy the following conditions:
1. \( a, b, c \in \mathbb{N} \);
2. \( a + b + c = n \);
3. \( p \nmid \frac{n!}{a! \cdot b! \cdot c!} \).

How many ordered triples \((a, b, c)\) are there in the set \( S_{n} \)?

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