The owner of a small brewery is using Holt$-$Winter$$s method to forecast his quarterly beer

sales. He has been smoothing constants of $=0\mathrm{.}2,=0\mathrm{.}2,$ and $=0\mathrm{.}2.$ He has currently

obtained the following values of the various level, trend, and seasonal factors: $"$# $=$

$120,"$# $=14,"$# $=1\mathrm{.}2,$$ $=1\mathrm{.}1,\%=0\mathrm{.}8,$ and $$& $=0\mathrm{.}9.$

$(1)$ Determine the forecast for the beer sales in quarter$11.$

$(2)$ Suppose that the actual sales turn out to be $128$ in quarter $11.$ Find $""$ and $"",$ and find

the updated values of the seasonal factors. Also determine the forecast made at the end

of quarter $11$ for quarter $13.$