Galaxy Co$.$ sells virtual reality $($VR$)$ goggles targeted to customers who like to play video games. Galaxy procures each pair of goggles for $$150$ from its supplier and sells each pair of goggles for $$300.$ Monthly demand for the VR goggles is a normal random variable with a mean of $180$ units and a standard deviation of $45$ units. At the beginning of each month, Galaxy orders enough goggles from its supplier to bring the inventory level up to $160$ goggles. If the monthly demand is less than $160,$ Galaxy pays $$20$ per pair of goggles that remains in inventory at the end of the month. If the monthly demand exceeds $160,$ Galaxy sells only the $160$ pairs of goggles in stock. Galaxy assigns a shortage cost of $$40$ for each unit of demand that is unsatisfied to represent a loss$-$of$-$goodwill among its customers. Management would like to use a simulation model to analyze this situation.

What is the average monthly profit resulting from its policy of stocking $160$ routers at the beginning of each month? Round your answer to the nearest dollar.

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What is the proportion of months in which demand is completely satisfied? Round your answer to the nearest whole number.

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Use the simulation model to compare the profitability of monthly replenishment levels of $105,160,$ and $195$ routers. Which monthly replenishment level maximizes profitability?

Use the corresponding $95\%$ confidence intervals on the average profit to make your comparison. Round your answers to the nearest dollar.

Lower Bound: $\_\_\_\_\_\_\_\_$

Upper Bound: $\_\_\_\_\_\_\_\_$