Albert and Arjun are situated on a large hexagon. This hexagon has a candle at each of the vertices. Albert wishes to light all of these candles at the same time; however, each candle is special, and takes a different amount of time to light up. The first candle takes $1$ second to light up, the second takes $2$ seconds, and so on (these candles are consecutively placed). Albert and Arjun decide to split up the work of turning each of these on, and can move from one vertex to an adjacent one in $1$ second. To make sure they light the candles in the right order, they call out to each other to notify of the lighting of a candle. However, the speed of sound is slow in this dimension, and also takes $1$ second to propagate along an edge. Albert begins at vertex $1$ and Arjun begins at vertex $4$. What is the minimum time for the hexagon to be fully lit?