Cow Cave Adventure

Few people know that cows love to explore caves. In a cave there are $N$ ($1 \leq N \leq 100$) chambers connected by $M$ ($1 \leq M \leq 1000$) bidirectional corridors. Each pair of chambers is connected by at most one corridor. Additionally, there are $K$ ($1 \leq K \leq 14$) chambers that contain a bundle of hay. When Betsy the cow eats one bundle of hay, her weight index increases by $1$.

Betsy starts her adventure at chamber $1$ with a weight index of $0$. However, every corridor has a width threshold. If her current weight index exceeds the threshold of a corridor (i.e. if her weight index is greater than the threshold), she will get stuck. After exploring the cave, she must return to chamber $1$.

Determine the maximum number of hay bundles Betsy can eat. Note that when passing through a chamber with hay, she does not have to eat the hay immediately.

inputFormat

The first line contains three integers $N$, $M$, and $K$.

The next $M$ lines each contain three integers $u$, $v$, and $w$, indicating that there is a bidirectional corridor between chambers $u$ and $v$ with a width threshold $w$. (A corridor can be traversed if and only if the current weight index is less than or equal to its threshold.)

The following $K$ lines each contain one integer representing a chamber that contains a bundle of hay. It is guaranteed that these chambers are distinct.

outputFormat

Output a single integer representing the maximum number of hay bundles Betsy can eat while starting and ending at chamber $1$.

sample

4 4 2
1 2 3
2 3 1
3 4 1
4 1 1
2
3

1