$($Circulation$)$ The Galactic Council has tasked you with optimizing the distribution of rare crystals among research facilities on various planets. There are n types of crystals and m research facilities. Each crystal type must be allocated to exactly one facility. Facilities can accommodate between $1$ to $10$ different crystal types. Each crystal type has a list of compatible facilities it can be sent to$.$ Every facility must receive at least one crystal type for research. Design a polynomial$-$time algorithm to determine if a valid distribution of crystals to facilities is possible, satisfying all constraints. Your solution should also specify which crystal type is assigned to which facility. $($a$)$ Describe how to construct a flow circulation to model this problem. $(2,5)($b$)$ Make a claim regarding a valid distribution of this problem. Prove the claim in both directions.