Trucking Troubles

You sell trucks that can carry trucks that can carry trucks, so your trucks are extremely heavy. To deliver one, you must drive it across a wide, wet region, and because it is wet you have to cross bridges along the way.

The region has $c$ cities, numbered from $1$ to $c$. Between some pairs of cities there is a road, and every road carries a bridge — but not every pair of cities is connected by a direct road. Each bridge has a maximum weight capacity, an integer from $0$ to $100\,000$; a truck may cross a bridge only if the truck's weight does not exceed that capacity.

Some cities are destination cities, where customers are eager to see your truck. You start at city $1$ (which is never a destination city) and must visit all $d$ destination cities, in any order. Because you drive a single truck, its weight is fixed for the whole journey.

Determine the maximum truck weight for which you can start at city $1$ and still reach every destination city, using only bridges that can support that weight.

The first line contains three positive integers $c$, $r$, and $d$: the number of cities, the number of roads, and the number of destination cities. There are at most $10\,000$ cities and at most $100\,000$ roads.

Each of the next $r$ lines contains three integers $x\ y\ w$, meaning there is a road between city $x$ and city $y$ whose bridge has a maximum weight capacity of $w$.

Each of the next $d$ lines contains one destination city. There is at least one destination city, and city $1$ is never a destination.

Print a single integer: the largest weight that can be driven from city $1$ through all $d$ destination cities.