There are \(n\) pirates \(\left(P_{n}, P_{n-1}, \cdots P_1 \right) \) with the strict order of superiority \(\left(P_{n} > P_{n-1} > \cdots > P_1 \right) \).

They find 100 gold coins. They must distribute the coins among themselves according to the following rules:

- The superior-most surviving pirate proposes a distribution.
- The pirates, including the proposer, then vote on whether to accept this distribution.
- In case of a tie vote the proposer has the casting vote.
- If the distribution is accepted, the coins are disbursed and the game ends. If not, the proposer is thrown overboard from the pirate ship and dies (leaving behind \((n-1)\) pirates and their original ordering), and the next most superior pirate makes a new proposal to begin the game again.

Every pirate knows that every other pirates is perfectly *rational*, which means:

- They will always be able to deduce something logically deducible.
- Their first preference is that they want to survive.
- Their second preference is that they want to maximize the number of coins they receive.
- Other things remaining same, they prefer to kill a pirate rather than having him alive.

What is the minimum value of \(n\) such that the superior-most pirate can propose no distribution of coins that will let him survive?

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