A production line has three workstations. At station $1,$ there is one machine, which takes $2$ minutes to process one unit. The unit then passes to station $2,$ which has two machines in parallel. The unit must be processed on just one of the machines at station $2,$ and each machine at station $2$ takes $5$ minutes to process one unit. The unit then passes to station $3,$ which has $6$ machines in parallel. The unit must be processed on just one of the machines at station $3,$ and each machine at station $3$ takes $14$ minutes to process one unit. What is the bottleneck rate in units per hour?