There are $N$ rooms numbered $1$, $2$, $\ldots$, $N$, each with one person living in it, and $M$ corridors connecting two different rooms. The $i$\-th corridor connects room $U_i$ and room $V_i$ with a length of $W_i$. One day (we call this day $0$), the $K$ people living in rooms $A_1, A_2, \ldots, A_K$ got (newly) infected with a virus. Furthermore, on the $i$\-th of the following $D$ days $(1\leq i\leq D)$, the infection spread as follows. > People who were infected at the end of the night of day $(i-1)$ remained infected at the end of the night of day $i$. > For those who were not infected, they were newly infected if and only if they were living in a room within a distance of $X_i$ from at least one room where an infected person was living at the end of the night of day $(i-1)$. Here, the distance between rooms $P$ and $Q$ is defined as the minimum possible sum of the lengths of the corridors when moving from room $P$ to room $Q$ using only corridors. If it is impossible to move from room $P$ to room $Q$ using only corridors, the distance is set to $10^{100}$. For each $i$ ($1\leq i\leq N$), print the day on which the person living in room $i$ was newly infected. If they were not infected by the end of the night of day $D$, print $-1$.

The input is given from Standard Input in the following format: >$N$ $M$\ >$U_1$ $V_1$ $W_1$\ >$U_2$ $V_2$ $W_2$\ >$\vdots$\ >$U_M$ $V_M$ $W_M$\ >$K$\ >$A_1$ $A_2$ $\ldots$ $A_K$\ >$D$\ >$X_1$ $X_2$ $\ldots$ $X_D$ #### Constraints - $1 \leq N\leq 3\times 10^5$ - $0 \leq M\leq 3\times 10^5$ - $1 \leq U_i \lt V_i \leq N$ - All $(U_i,V_i)$ are different. - $1 \leq W_i \leq 10^9$ - $1 \leq K\leq N$ - $1 \leq A_1 \lt A_2 \lt \cdots \lt A_K \leq N$ - $1 \leq D\leq 3\times 10^5$ - $1 \leq X_i \leq 10^9$ - All input values are integers.

Print $N$ lines. The $i$\-th line $(1\leq i\leq N)$ should contain the day on which the person living in room $i$ was newly infected.

***For Sample #1:*** The infection spreads as follows. - On the night of day $0$, the person living in room $1$ gets infected. - The distances between room $1$ and rooms $2,3,4$ are $2,3,5$, respectively. Thus, since $X_1=3$, the people living in rooms $2$ and $3$ are newly infected on the night of day $1$. - The distance between rooms $3$ and $4$ is $2$. Thus, since $X_2=3$, the person living in room $4$ also gets infected on the night of day $2$. Therefore, the people living in rooms $1,2,3,4$ were newly infected on days $0,1,1,2$, respectively, so print $0,1,1,2$ in this order on separate lines. ***For Sample #3:*** Note that it is not always possible to move between any two rooms using only corridors.

AtCoder [ABC307F]