
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 a natural number and \( p \) a prime number.

Show using Legendre's formula (Exercise 8)

\[ 
v_{p}\left(\binom{2n}{n}\right) = \sum_{k=1}^{\left\lfloor\frac{\ln (2n)}{\ln p}\right\rfloor}\left(\left\lfloor\frac{2n}{p^{k}}\right\rfloor-2\left\lfloor\frac{n}{p^{k}}\right\rfloor\right)
\]

and then, using Exercise 5, deduce that 

\[
v_{p}\left(\binom{2n}{n}\right) \leqslant \left\lfloor\frac{\ln (2n)}{\ln p}\right\rfloor 
\]

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