$($a$)$ Suppose that a hash file with a simple hash function, i$.$e$.$ Age mod $9,$ is used for the table instead of indexed file. Suppose also that one bucket can hold two records of the table. Demonstrate the working of the hash file in terms of initial data loading and the subsequent retrieval. Consider a solution for potential collisions.

$($b$)$ Rather than hash file, demonstrate how to create and use a B$+$ tree index for fast search upon Age attribute.