Exercise $1($FTRL on hard OLO problems$).$

Recall the difficult online linear optimization $($OLO$)$ problem on which FTL fails:

The action space is $A=[-1,1],$

the environmental data sequences is $z_t=\{\begin{array}{cc}-\frac{1}{2},& t=1,\\ 1,& tis\text{e}\text{v}\text{e}\text{n}\text{,}& \\ -1,& tis\text{o}\text{d}\text{d}\text{,}& \end{array}$

and the loss function is $l(a,z)=a^{TT}z.$

Run FTRL for this problem setting with the optimal choice of $.$

Solve showing all mathematical steps.