FTOUR2 - Free tour II

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After the success of 2nd anniversary (take a look at problem FTOUR for more details), this 3rd year, Travel Agent SPOJ goes on with another discount tour.

The tour will be held on ICPC island, a miraculous one on the Pacific Ocean. We list N places (indexed from 1 to N) where the visitors can have a trip. Each road connecting them has an interest value, and this value can be negative (if there is nothing interesting to view there). Simply, these N places along with the roads connecting them form a tree structure. We will choose two places as the departure and destination of the tour.

Since September is the festival season of local inhabitants, some places are extremely crowded (we call them crowded places). Therefore, the organizer of the excursion hopes the tour will visit at most K crowded places (too tiring to visit many of them) and of course, the total number of interesting value should be maximum.

Briefly, you are given a map of N places, an integer K, and M id numbers of crowded place. Please help us to find the optimal tour. Note that we can visit each place only once (or our customers easily feel bored), also the departure and destination places don't need to be different.

Input

There is exactly one case. First one line, containing 3 integers N K M, with 1 <= N <= 200000, 0 <= K <= M, 0 <= M <= N.

Next M lines, each line includes an id number of a crowded place.

The last (N - 1) lines describe (N - 1) two-way roads connected N places, form a b i, with a, b is the id of 2 places, and i is its interest value (-10000 <= i <= 10000).

Output

Only one number, the maximum total interest value we can obtain.

Example

Input:
8 2 3
3
5
7
1 3 1
2 3 10
3 4 -2
4 5 -1
5 7 6
5 6 5
4 8 3


Output:
12

Explanation

We choose 2 and 6 as the departure and destination place, so the tour will be 2 -> 3 -> 4 -> 5 -> 6, total interest value = 10 + (-2) + (-1) + 5 = 12
* Added some unofficial cases


hide comments
shimomire: 2016-11-13 15:04:45

What is the output of the following input?
I think that there does not exist any valid path.

1 0 1
1

plosive: 2016-10-14 15:04:52

how solve this with use Boundary partition´╝č

w703710691d: 2015-09-04 15:55:24

why I always get a re?

lu: 2015-06-01 17:08:24

the result turned to TLE QAQ

lu: 2015-06-01 16:03:01

my program passed the following tests but it got WA.I don't know what's wrong

kid: 2015-01-07 15:17:10

I chanllenged many codes using the following input
2 0 0
1 2 1
pay attention to the initialization

yzx: 2012-09-23 07:05:00

answer:6

Exia_cai: 2012-09-14 08:07:12

here is a test:
5 3 4
2
3
4
5
1 2 1
2 3 3
1 4 1
1 5 2

谢良: 2011-06-09 12:31:58

300

Zhao Tao: 2010-04-27 03:13:28

The test data of this problem is somewhat weak. I successfully chanllenged a code which was accepted using the following input:
7 5 6
2
3
4
5
6
7
1 7 100
1 5 100
5 6 100
1 2 1
2 3 1
3 4 1


Added by:Thanh-Vy Hua
Date:2007-09-28
Time limit:0.100s-0.443s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO NODEJS PERL6 VB.NET
Resource:Adapted from Preslav Le's problem, first used in Bulgarian OI 07