To draw a project network, we need to identify the burst activities, merge activities, and the critical path.

Burst activities are activities that have multiple successors but only one predecessor. In this case, the burst activity is activity B $($Excavate Site$).$

Merge activities are activities that have multiple predecessors but only one successor. In this case, there are no merge activities.

The critical path is the longest path through the network, which determines the minimum project duration. To find the critical path, we need to calculate the total duration for each path and identify the path with the longest duration.

The project network for the given information is as follows:

A $->$ B $->$ C $->$ E

$\backslash ->$ D $->$ E

To find the critical path, we calculate the total duration for each path:

Path $1$: A $->$ B $->$ C $->$ E

Total duration $=2+4+3+3=12$ days

Path $2$: A $->$ B $->$ D $->$ E

Total duration $=2+4+2+3=11$ days

Therefore, the critical path is Path $1$: A $->$ B $->$ C $->$ E with a total duration of $12$ days.