Description:
Michael and Brian are stuck in a hotel with $$$n$$$ rooms, numbered from $$$1$$$ to $$$n$$$, and need to find each other. But this hotel's doors are all locked and the only way of getting around is by using the teleporters in each room. Room $$$i$$$ has a teleporter that will take you to room $$$a_i$$$ (it might be that $$$a_i = i$$$). But they don't know the values of $$$a_1,a_2, \dots, a_n$$$.
Instead, they can call up the front desk to ask queries. In one query, they give a room $$$u$$$, a positive integer $$$k$$$, and a set of rooms $$$S$$$. The hotel concierge answers whether a person starting in room $$$u$$$, and using the teleporters $$$k$$$ times, ends up in a room in $$$S$$$.
Brian is in room $$$1$$$. Michael wants to know the set $$$A$$$ of rooms so that if he starts in one of those rooms they can use the teleporters to meet up. He can ask at most $$$2000$$$ queries.
The values $$$a_1, a_2, \dots, a_n$$$ are fixed before the start of the interaction and do not depend on your queries. In other words, the interactor is not adaptive.
Input Format:
The input contains a single integer $$$n$$$ ($$$2 \leq n \leq 500$$$).
Output Format:
None
Note:
In the sample test, there are $$$n=5$$$ rooms and the (hidden) array describing the behavior of the teleporters is $$$[1, 2, 1, 3, 2]$$$.
- The first query asks whether starting from room number $$$a=3$$$, and using the teleporters $$$5$$$ times, one ends up in one of the two rooms $$$S=\{2, 3\}$$$. This action results in ending up in the room $$$1$$$, so the answer is $$$0$$$.
- The second query asks whether starting from room number $$$a=2$$$, and using the teleporters $$$5$$$ times, one ends up in one of the two rooms $$$S=\{2, 3\}$$$. This action results in ending up in the room $$$2$$$, so the answer is $$$1$$$.