In a room, there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you turn on the $n$-th switch, the $n$-th bulb will light up. There is a group of men who are operating the switches according to the following rule: 

- They enter one by one and start flipping the switches starting from the first switch until they have to turn on a bulb. 
- As soon as they turn a bulb on, they leave the room.

For example:
- The first person enters, turns the first switch on, and leaves.
- The second person enters, sees the first switch on, turns it off, and then lights the second bulb.
- The third person enters, finds the first switch off, turns it on, and leaves the room.
- The fourth person enters, switches off the first and second bulbs, and switches on the third.

This process continues in this manner. Finally, we find out that the first 10 bulbs are off and the 11th bulb is on. How many people were involved in the entire process?

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