$3.$ Consider a semi$-$automated packaging machine. A worker loads the products into the machine, the machine packs the product, and the worker unloads the packed product.

$-$Loading takes $2$ minutes.

$-$Packaging takes $5$ minutes.

$-$Unloading takes $1$ minute.

$-$The machine can only pack one product at a time.

$-$The worker must finish loading or unloading when they start it$.$ They cannot leave these tasks unfinished.

$-$The packing starts automatically as soon as the product is loaded.

$-$The unloading can be done anytime after the packing is finished. The packed product can wait in the machine until the worker unloads it$.$ The worker must unload the packed product before loading a new product into the machine.

a$.$ Calculate the worker's maximum utilization factor when one worker operates one machine. Note that the worker is only busy during loading and unloading and is idle while the machine packs the product.

b$.$ What is the maximum number of machines a single worker can operate? Draw a timing schedule for $100\%$ utilization factor. $($We do not want the worker to be in idle mode$)$