Problem Statement

Takahashi is performing a terrain simulation of a mountainous area.

The mountainous area has N locations, each numbered from 1 to N. Location i has an elevation of H_i and initially holds W_i liters of rainwater.

The locations are connected by M channels. The j-th channel bidirectionally connects location U_j and location V_j. Water flows only from locations with higher elevation to locations with lower elevation. That is, if two locations connected by a channel have the same elevation, water does not flow through that channel.

For a location v, among the locations directly connected to v by a channel, those with elevation strictly lower than v are called the downstream neighbors of v.

Takahashi has installed dams at K locations (if K = 0, no dams are installed). The locations where dams are installed are numbered S_1, S_2, \ldots, S_K. At a location where a dam is installed, water may flow in from other locations through channels, but no water ever flows out from that location.

The water outflow from each location is processed according to the following procedure:

1. Sort all locations in decreasing order of elevation. Since water does not flow between locations of the same elevation, the ordering among locations with the same elevation is arbitrary (the result does not depend on the processing order).
2. Process each location one by one in this order. Let v be the location being processed, and let w liters be the amount of water currently accumulated at v.

• If a dam is installed at v: Do nothing. All w liters of water remain at v.
• If no dam is installed at v, and there are d downstream neighbors (d \geq 1): All w liters of water at v flow out. The outflowing water is distributed equally among the d downstream neighbors, with each downstream neighbor receiving \frac{w}{d} liters of water (water flows into a downstream neighbor even if a dam is installed there). After the outflow, the water amount at v becomes 0.
• If no dam is installed at v, and there are no downstream neighbors (including isolated locations with no connected channels, or locations where all connected locations have elevation greater than or equal to v): The water remains at v as is.

When water flows into a downstream neighbor during the outflow processing of some location, that water is immediately accumulated at the downstream neighbor and is handled collectively when that downstream neighbor itself is processed.

After all outflow processing is complete, determine the amount of water remaining at each location.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 0 \leq M \leq 2 \times 10^5
• 0 \leq K \leq N
• 1 \leq H_i \leq 10^9 (1 \leq i \leq N)
• 0 \leq W_i \leq 10^9 (1 \leq i \leq N)
• 1 \leq U_j < V_j \leq N (1 \leq j \leq M)
• There is at most one channel connecting the same pair of locations (no multi-edges)
• 1 \leq S_k \leq N (1 \leq k \leq K)
• S_1, S_2, \ldots, S_K are all distinct
• All input values are integers

Sample Input 1

4 4 0
10 20 5 5
0 12 0 0
1 2
1 3
2 3
2 4

Sample Output 1

0.0000000000 0.0000000000 8.0000000000 4.0000000000

Sample Input 2

3 2 1
30 20 10
6 0 0
1 2
2 3
2

Sample Output 2

0.0000000000 6.0000000000 0.0000000000

Sample Input 3

6 7 1
50 40 40 30 20 10
18 6 0 0 0 0
1 2
1 3
2 4
2 5
3 5
4 6
5 6
5

Sample Output 3

0.0000000000 0.0000000000 0.0000000000 0.0000000000 16.5000000000 7.5000000000

Sample Input 4

8 10 2
100 80 80 60 50 50 30 10
24 0 0 0 12 0 0 0
1 2
1 3
2 4
3 4
3 5
4 6
4 7
5 7
6 8
7 8
4 6

Sample Output 4

0.0000000000 0.0000000000 0.0000000000 18.0000000000 0.0000000000 0.0000000000 0.0000000000 18.0000000000

Sample Input 5

1 0 0
1000000000
999999999

Sample Output 5

999999999.0000000000