Jam

E: Jam

problem

There are N cities in a country, numbered 1, \ 2, \ ..., \ N. These cities are connected in both directions by M roads, and the i-th road allows you to travel between the cities u_i and v_i in time t_i. Also, any two cities can be reached by using several roads.

Bread is sold in each town, and the deliciousness of bread sold in town i is P_i.

In addition, there are K flavors of jam in this country, and in town i you can buy delicious J_i jams at taste c_i.

Homu-chan, who lives in Town 1, decided to go out to buy bread and jam one by one. Homu-chan travels through several cities, buys bread and jam, and returns to city 1. More precisely, go to city u to buy bread, select city v to buy jam, and return to city 1. At this time, u = v, u = 1, and v = 1 can be used.

The happiness of Homu-chan after shopping is "the total of the deliciousness of the bread and jam I bought"-"the time it took to move". For each of the K types of jam, calculate the maximum possible happiness when going to buy jam and bread of that taste.

Input format

NMK P_1 P_2 ... P_N c_1 c_2 ... c_N J_1 J_2 ... J_N u_1 v_1 t_1 u_2 v_2 t_2 :: u_M v_M t_M

Constraint

• 1 \ leq N \ leq 10 ^ 5
• 1 \ leq M \ leq 2 \ times 10 ^ 5
• 1 \ leq K \ leq N
• 1 \ leq P_i, J_i \ leq 10 ^ 9
• 1 \ leq c_i \ leq K
• 1 \ leq u_i, v_i \ leq N
• 1 \ leq t_i \ leq 10 ^ 9
• For all i \ (1 \ leq i \ leq K), there exists j \ (1 \ leq j \ leq N) such that c_j = i.
• Given graphs are concatenated graphs that do not include multiple edges or self-loops.
• All inputs are given as integers

Output format

Please output K line. On line i, print the maximum happiness when you go to buy bread and taste i jam.

Input example 1

4 4 3 3 6 1 6 1 1 2 3 6 1 5 5 1 2 1 2 3 1 1 3 1 1 4 1

Output example 1

Ten 8 9

• When buying taste 1 jam
• If you buy jam in city 1, move to city 2, buy bread, and come back to city 1, your happiness will be 6 + 6-2 = 10, which is optimal.
• When buying taste 2 jam
• If you move to City 2 to buy bread, move to City 3 to buy jam, and return to City 1, your happiness will be 6 + 5-3 = 8.
• When buying taste 3 jam
• If you go to City 4 and buy both bread and jam and come back to City 1, your happiness will be 6 + 5-2 = 9.

Input example 2

2 1 2 1 1 1 2 1 1 1 2 1000000000

Output example 2

2 -1999999998

• Homu-chan seems to be dissatisfied because the distance is too far compared to the deliciousness of bread and jam.

Input example 3

6 8 3 31 41 59 26 53 58 1 2 3 1 2 3 27 18 28 18 28 46 one two Three 2 3 7 2 4 8 3 6 9 3 5 3 4 5 3 1 5 15 4 6 7

Output example 3

66 61 68

Example

Input

4 4 3 3 6 1 6 1 1 2 3 6 1 5 5 1 2 1 2 3 1 1 3 1 1 4 1

Output

10 8 9

inputFormat

Input format

NMK P_1 P_2 ... P_N c_1 c_2 ... c_N J_1 J_2 ... J_N u_1 v_1 t_1 u_2 v_2 t_2 :: u_M v_M t_M

Constraint

• 1 \ leq N \ leq 10 ^ 5
• 1 \ leq M \ leq 2 \ times 10 ^ 5
• 1 \ leq K \ leq N
• 1 \ leq P_i, J_i \ leq 10 ^ 9
• 1 \ leq c_i \ leq K
• 1 \ leq u_i, v_i \ leq N
• 1 \ leq t_i \ leq 10 ^ 9
• For all i \ (1 \ leq i \ leq K), there exists j \ (1 \ leq j \ leq N) such that c_j = i.
• Given graphs are concatenated graphs that do not include multiple edges or self-loops.
• All inputs are given as integers

outputFormat

Output format

Please output K line. On line i, print the maximum happiness when you go to buy bread and taste i jam.

Input example 1

4 4 3 3 6 1 6 1 1 2 3 6 1 5 5 1 2 1 2 3 1 1 3 1 1 4 1

Output example 1

Ten 8 9

• When buying taste 1 jam
• If you buy jam in city 1, move to city 2, buy bread, and come back to city 1, your happiness will be 6 + 6-2 = 10, which is optimal.
• When buying taste 2 jam
• If you move to City 2 to buy bread, move to City 3 to buy jam, and return to City 1, your happiness will be 6 + 5-3 = 8.
• When buying taste 3 jam
• If you go to City 4 and buy both bread and jam and come back to City 1, your happiness will be 6 + 5-2 = 9.

Input example 2

2 1 2 1 1 1 2 1 1 1 2 1000000000

Output example 2

2 -1999999998

• Homu-chan seems to be dissatisfied because the distance is too far compared to the deliciousness of bread and jam.

Input example 3

6 8 3 31 41 59 26 53 58 1 2 3 1 2 3 27 18 28 18 28 46 one two Three 2 3 7 2 4 8 3 6 9 3 5 3 4 5 3 1 5 15 4 6 7

Output example 3

66 61 68

Example

Input

4 4 3 3 6 1 6 1 1 2 3 6 1 5 5 1 2 1 2 3 1 1 3 1 1 4 1

Output

10 8 9

样例

4 4 3
3 6 1 6
1 1 2 3
6 1 5 5
1 2 1
2 3 1
1 3 1
1 4 1

10
8
9

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