C. Doe Graphs
time limit per test
3 seconds
memory limit per test
256 megabytes
standard input
standard output

John Doe decided that some mathematical object must be named after him. So he invented the Doe graphs. The Doe graphs are a family of undirected graphs, each of them is characterized by a single non-negative number — its order.

We'll denote a graph of order k as D(k), and we'll denote the number of vertices in the graph D(k) as |D(k)|. Then let's define the Doe graphs as follows:

  • D(0) consists of a single vertex, that has number 1.
  • D(1) consists of two vertices with numbers 1 and 2, connected by an edge.
  • D(n) for n ≥ 2 is obtained from graphs D(n - 1) and D(n - 2). D(n - 1) and D(n - 2) are joined in one graph, at that numbers of all vertices of graph D(n - 2) increase by |D(n - 1)| (for example, vertex number 1 of graph D(n - 2) becomes vertex number 1 + |D(n - 1)|). After that two edges are added to the graph: the first one goes between vertices with numbers |D(n - 1)| and |D(n - 1)| + 1, the second one goes between vertices with numbers |D(n - 1)| + 1 and 1. Note that the definition of graph D(n) implies, that D(n) is a connected graph, its vertices are numbered from 1 to |D(n)|.
The picture shows the Doe graphs of order 1, 2, 3 and 4, from left to right.

John thinks that Doe graphs are that great because for them exists a polynomial algorithm for the search of Hamiltonian path. However, your task is to answer queries of finding the shortest-length path between the vertices ai and bi in the graph D(n).

A path between a pair of vertices u and v in the graph is a sequence of vertices x1, x2, ..., xk (k > 1) such, that x1 = u, xk = v, and for any i (i < k) vertices xi and xi + 1 are connected by a graph edge. The length of path x1, x2, ..., xk is number (k - 1).


The first line contains two integers t and n (1 ≤ t ≤ 105; 1 ≤ n ≤ 103) — the number of queries and the order of the given graph. The i-th of the next t lines contains two integers ai and bi (1 ≤ ai, bi ≤ 1016, ai ≠ bi) — numbers of two vertices in the i-th query. It is guaranteed that ai, bi ≤ |D(n)|.

Please, do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use cin, cout streams or the %I64d specifier.


For each query print a single integer on a single line — the length of the shortest path between vertices ai and bi. Print the answers to the queries in the order, in which the queries are given in the input.

10 5
1 2
1 3
1 4
1 5
2 3
2 4
2 5
3 4
3 5
4 5