A mail$-$order house uses $15,750$ boxes a year. Carrying costs are $45$ cents per box a year, and ordering costs are $$92.$ The following price schedule applies.

Number of Boxes Price per Box

$1,000$ to $1,999$ $$1\mathrm{.}40$

$2,000$ to $4,9991\mathrm{.}30$

$5,000$ to $9,9991\mathrm{.}20$

$10,000$ or more $1\mathrm{.}15$

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a$.$ Determine the optimal order quantity. $($Round your answer to the nearest whole number.$)$

b$.$ Determine the number of orders per year. $($Round your answer to $2$ decimal places.$)$