$3.$ A fulfillment center processes online orders that require workers to pick items from shelves and move them to the packing zone. On average, it takes a worker $45$ minutes to complete a full pick$-$pack cycle, including walking, handling, and occasional scanning delays. Additionally, workers take a $15-$minute break after every $3$ cycles. $($a$)$ The manager wants to dispatch one worker into the picking area every $5$ minutes during a $16-$hour operational window to maintain continuous throughput. Estimate the minimum number of workers needed to support this dispatch schedule. $($b$)$ A supervisor proposes halting order picking between $2$ a$.$m$.$ and $6$ a$.$m$.$ for restocking and maintenance. Will this off$-$hour break lead to a reduction in the number of workers needed for maintaining the same dispatch frequency during active hours? Explain your reasoning.