The Busy Busy Robot

It's the year 2050, and Apple just released their iPhone 29. Trying to ship their "new" product to as many cities as possible, a robot is designed to organize the phones into boxes, each box being capable of holding up to 100 phones. Every night the robot is tasked with filling up 35 boxes labeled 1 to 35. To fill the boxes, the robot chooses a random box, places a phone in it, and repeats these two steps until each box is full. Also, each box is equally likely to be chosen.

After randomly placing 25 phones, the probability that the 26\(^\text{th}\) phone is placed in a box with no other phones in it is \(X\).

Find \(\displaystyle \lceil 10000X \rceil\).

You may use computer assistance for the final step of your calculation.

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