
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
Consider the factorial and binomial coefficient problem related to exponents. Let's first state a lemma.

Lemma: \( v_{p}(n!)=\frac{n-\tau_{p}(n)}{p-1} \).

Proof: Let \( n = \sum_{i=0}^{s} n_{i} p^{i} \) where \( 0 \leq n_{i} \leq p-1 \) and \( n_{s} \neq 0 \).

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