Let ~T=(V, E, W)~ be an acyclic and connected undirected graph (also known as an unrooted tree), each edge has a positive integer weight, we call ~T~ a tree network, where ~V, E~ Respectively represent the collection of nodes and edges, ~W~ represents the collection of the length of each side, and let ~T~ have n nodes.

Definitions

• Path: There is a unique simple path between any two nodes ~a~ and ~b~ in the tree network. Let ~d(a, b)~ represent the length of the path with a and b as the endpoints, which is the sum of the lengths of the sides on the path. We call ~d(a,b)~ the distance between two nodes ~a,b~. The distance from a point ~v~ to a path P is the distance from that point to the nearest node on P: ~d(v, P)=\min\{d(v, u), \text{u is the node on the path P}\}~

• Diameter of the tree network: The longest path in the tree network is called the diameter of the tree network. For a given treenet ~T~, the diameter is not necessarily unique, But it can be proved that the midpoint of each diameter (not necessarily exactly a certain node, it may be inside a certain side) is unique, and we call this point the center of the tree network.

• Eccentric distance ~ECC(F)~: the distance from the node farthest from the path ~F~ in the tree network T to the path ~F~, that is ~ECC(F) = \max\{d(v, F), v \in V\}~

Task

For a given tree network ~T=(V, E, W)~ and a non-negative integer ~s~, find a path ~F~, which is a path on a certain diameter (both ends of the path are nodes in the tree network) , whose length does not exceed ~s~ (can be equal to ~s~), so that the eccentricity ~ECC(F)~ is the smallest. We call this path the core of the tree network ~T=(V,E,W)~. When necessary, ~F~ can degenerate to a node. Generally speaking, under the above definition, there is not necessarily only one core, but the minimum eccentricity is unique.

The figure below shows an example of a tree network. In the figure, ~A-B~ and ~A-C~ are two diameters with a length of 20. Point ~W~ is the center of the treenet, and the length of the ~EF~ edge is ~5~. If ~s=11~ is specified, the core of the tree network is path ~DEFG~ (or path ~DEF~), and the eccentricity is ~8~. If ~s=0~ (or ~s=1~ or ~s=2~), the core of the tree network is node ~F~ with an eccentricity of ~12~.

Input Specification

• Line ~1~, two positive integers ~n~ and ~s~, separated by a space. ~n~ is the number of nodes in the tree network, and ~s~ is the upper bound of the length of the core of the tree network. Let the node numbers be ~1, 2, \ldots n~.
• Lines ~2~ to ~n~ each contains 3 positive integers separated by spaces, indicating the two endpoint numbers and length of each edge in turn. For example, 2 4 7 means that the length of the edge connecting nodes ~2~ and ~4~ is ~7~.

It's guaranteed that the input forms a valid tree.

Output Specification

One non-negative numbers, the minimum eccentricity under this condition.

Sample Input 1

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

Sample Output 1

5

Sample Input 2

8 6 
1 3 2 
2 3 2 
3 4 6 
4 5 3 
4 6 4 
4 7 2 
7 8 3

Sample Output 2

5

Constraints

• ~32\%~ of the test cases satisfy ~5 \leq n \leq 15~.
• ~56\%~ of the test cases satisfy ~5 \leq n \leq 80~.
• ~80\%~ of the test cases satisfy ~5 \leq n \leq 300~, ~0 \leq s \leq 1\,000~.
• ~90\%~ of the test cases satisfy ~5 \leq n \leq 5\times 10^5~, ~0 \leq s < 2^{31}~, and all lengths are positive integers not exceeding ~1\,000~.
• ~100\%~ of the test cases satisfy ~5 \leq n \leq 2\times 10^6~, ~0 \leq s < 2^{31}~, and all lengths are positive integers not exceeding ~1\,000~.