Using the simplifying assumptions of Example $8\mathrm{.}7,$ suppose that there are three advertisers, A$,$ B$,$ and C$.$ There are three queries, x$,$ y$,$ and z$.$ Each advertiser has a budget of $2.$ Advertiser A bids only on x; B bids on x and y$,$ while C bids on x$,$ y$,$ and z$.$ Note that on the query sequence xxyyzz$,$ the optimum off$-$line algorithm would yield a revenue of $6,$ since all queries can be assigned.

$($a$)$ Show that the greedy algorithm will assign at least $4$ of these $6$ queries.

$($b$)$ Find another sequence of queries such that the greedy algorithm can assign as few as half the queries that the optimum off$-$line algorithm assigns on that sequence.