Q$.3[40$ pts$,20$ pts each$]$

Given an extendable hash index with the following contents. Each bucket can contain at most $4$ key values and the hash value of a search key $\backslash ($ x $\backslash )$ is the binary value of $\backslash ($ x $\backslash ).$ The least significant bit of a hash value is used during insertion.

$($a$)$ What is the minimum number of key$-$value insertions needed to increase the global depth of the bucket address table from $1$ to $3($i$.$e$.,$ expand the address table to a length of $8)?$ Provide one such sequence of insertions and indicate which insertions lead to an increase in the global depth of the bucket address table. Give the final contents of the index structure.

$($b$)$ What is the maximum number of key$-$value insertions needed to increase the global depth of the bucket address table from $1$ to $3?$ Provide one such sequence of insertions and indicate which insertions lead to an increase in the global depth of the bucket address table. Give the final contents of the index structure.