
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 and let the representation of the number \( n \) in the base \( p \) system be: 
\[ n = a_{k} p^{k} + a_{k-1} p^{k-1} + \ldots + a_{1} p^{1} + a_{0} \]

Find a formula expressing the exponent \(\alpha_{p}\), which is the highest power of \( p \) that divides \( n! \), in terms of \( n, p \), and the coefficients \( a_{k} \).

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