Ehab and a weird weight formula

You're given a tree consisting of n nodes. Every node u has a weight a_u. It is guaranteed that there is only one node with minimum weight in the tree. For every node u (except for the node with the minimum weight), it must have a neighbor v such that a_v<a_u. You should construct a tree to minimize the weight w calculated as follows:

• For every node u, deg_u ⋅ a_u is added to w (deg_u is the number of edges containing node u).
• For every edge { u,v }, ⌈ log_2(dist(u,v)) ⌉ ⋅ min(a_u,a_v) is added to w, where dist(u,v) is the number of edges in the path from u to v in the given tree.

Input

The first line contains the integer n (2 ≤ n ≤ 5 ⋅ 10^5), the number of nodes in the tree.

The second line contains n space-separated integers a_1, a_2, …, a_n (1 ≤ a_i ≤ 10^9), the weights of the nodes.

The next n-1 lines, each contains 2 space-separated integers u and v (1 ≤ u,v ≤ n) which means there's an edge between u and v.

Output

Output one integer, the minimum possible value for w.

Examples

Input

3 1 2 3 1 2 1 3

Output

7

Input

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

Output

40

Note

In the first sample, the tree itself minimizes the value of w.

In the second sample, the optimal tree is:

inputFormat

Input

The first line contains the integer n (2 ≤ n ≤ 5 ⋅ 10^5), the number of nodes in the tree.

The second line contains n space-separated integers a_1, a_2, …, a_n (1 ≤ a_i ≤ 10^9), the weights of the nodes.

The next n-1 lines, each contains 2 space-separated integers u and v (1 ≤ u,v ≤ n) which means there's an edge between u and v.

outputFormat

Output

Output one integer, the minimum possible value for w.

Examples

Input

3 1 2 3 1 2 1 3

Output

7

Input

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

Output

40

Note

In the first sample, the tree itself minimizes the value of w.

In the second sample, the optimal tree is:

样例

3
1 2 3
1 2
1 3

7