Zhang3 is playing a shooting game with Father. In the game there are two players trying to kill each other to win the game.
The game provides $n$ weapons, each has two properties: Damage and Delay. The $i^\mathrm{th}$ weapon has Damage $A_i$ and Delay $D_i$. When a player shoots with this weapon, his enemy's HP is reduced by $A_i$, then he must wait for $D_i$ ms before he can shoot again.
The game processes as follows:
1. Before the game starts, Zhang3 and Father choose a weapon respectively. Father always randomly chooses one of the $n$ weapons with equal probabilities. Each player can only use the chosen weapon during the game.
2. When the game starts, Zhang3 and Father have $100$ HP each. They make their first shot at the same time.
3. They keep shooting as quickly as possible. That means, a player shoots instantly whenever he can shoot, until the game ends.
4. When a player's HP is reduced to 0 or lower, he dies and the game ends. If the other player is still alive (i.e. has HP higher than 0), then the living player wins the game; otherwise (if the two players die at the same time), each player has $50\%$ probability to win the game.
Zhang3 wants to win the game. Please help her to choose a weapon so that the probability to win is maximized. Print the optimal probability.