Problem 1. Lineup Queries

USACO 2026 First Contest, Silver

There is a line of cows, initially (i.e. at time $t = 0$) containing only cow $0$ at position $0$ (here, a cow is at position $k$ if there are $k$ cows in front of it). At time $t$ for $t=1,2,3,\dots$, the cow at position 0 moves to position $\lfloor t/2\rfloor$, every cow in positions $1\dots \lfloor t/2\rfloor$ moves forward one position, and cow $t$ joins the line at the end of the line (position $t$).

Answer $Q$ ($1\le Q\le 10^5$) independent queries, each of one of the following types:

1. At what position is cow $c$ immediately after time $t$ ($0\le c\le t\le 10^{18}$)?
2. Which cow is at position $x$ immediately after time $t$ ($0\le x\le t\le 10^{18}$)?

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $Q$, the number of queries.

The next $Q$ lines each contain three integers specifying a query either of the form "1 c t" or "2 x t."

OUTPUT FORMAT (print output to the terminal / stdout):

Output the answer to each query on a separate line.

SAMPLE INPUT:

2
1 4 9
2 2 9

SAMPLE OUTPUT:

2
4

Lineups immediately after various times:

t = 0 | 0
t = 1 | 0 1
t = 2 | 1 0 2
t = 3 | 0 1 2 3
t = 4 | 1 2 0 3 4
t = 5 | 2 0 1 3 4 5
t = 6 | 0 1 3 2 4 5 6
t = 7 | 1 3 2 0 4 5 6 7
t = 8 | 3 2 0 4 1 5 6 7 8
t = 9 | 2 0 4 1 3 5 6 7 8 9

Immediately after $t=9$, the location of cow $4$ is $2$, and the cow located at position $2$ is $4$.

SAMPLE INPUT:

22
1 0 9
1 1 9
1 2 9
1 3 9
1 4 9
1 5 9
1 6 9
1 7 9
1 8 9
1 9 9
2 0 9
2 1 9
2 2 9
2 3 9
2 4 9
2 5 9
2 6 9
2 7 9
2 8 9
2 9 9
1 0 1000000000000000000
2 0 1000000000000000000

SAMPLE OUTPUT:

1
3
0
4
2
5
6
7
8
9
2
0
4
1
3
5
6
7
8
9
483992463350322770
148148148148148148

SCORING: Input 3: $Q\le 1000, t\le 100$Input 4: $t\le 5000$Inputs 5-8: All queries are of type 1.Inputs 9-12: All queries are of type 2.

Problem credits: Agastya Goel