$7\mathrm{.}4($Phased UCB $($i$))$ Fix a $1-$subgaussian k$-$armed bandit environment and a$$horizon n$.$ Consider the version of UCB that works in phases of exponentially$$increasing length of $1,2,4,....$ In each phase, the algorithm uses the action that$$would have been chosen by UCB at the beginning of the phase $($see Algorithm $4$below$).($a$)$ State and prove a bound on the regret for this version of UCB.