You have a directed acyclic graph G, consisting of n vertexes, numbered from 0 to n-1. The graph contains n edges numbered from 0 to n-1. An edge with number i connects vertexes i and (i+1)modn, and it can be directed in either direction (from i to (i+1)modn, or vise versa).
Operation xmody means taking the remainder after dividing number x by number y.
Let's call two vertexes u and v in graph G comparable if the graph contains a path either from u to v or from v to u. We'll assume that an antichain is a set of vertexes of graph G, where any two distinct vertexes are not comparable. The size of an antichain is the number of vertexes in the corresponding set. An antichain is maximum if the graph doesn't have antichains of a larger size.
Your task is to find the size of the maximum antichain in graph G.
Output
Print a single integer − the size of the maximum antichain of graph
G.
Note
Consider the first test sample. The graph's
G edges are:
0→1,
1→2,
0→2. We can choose the set of vertexes
[0] as the maximum antichain. We cannot choose an antichain of larger size.