In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows: 

1. The first contestant takes $\frac{1}{n}$ of the cake.
2. The second contestant takes $\frac{2}{n}$ of the remaining cake.
3. The third contestant takes $\frac{3}{n}$ of what remains after the first and second contestants have taken their portions.
4. This process continues until the last contestant, who takes all of the remaining cake.

Determine which competitor takes the largest piece of cake.

Let's think step by step and output the final answer within \boxed{}.
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