Bipartite Graphs and Matchings · Hard problem

Six tasks $1,\dots,6$ need machines from $\{a,b,c,d,e,f\}$: $1$ and $2$ can each use $a$ or $b$; $3$ and $4$ can each use $a$, $b$, or $c$; $5$ can use $d$ or $e$; $6$ can use $e$ or $f$. What is the maximum number of tasks that can run at the same time, one machine per task?