In an all$-$unit quantity discount context, price c$1$ is automatically dropped to c$2$ when the retailer buys $1000$ or more units at once. The retailer computes its economic order quantities for orders less $1000$ and for orders more than or equal to $1000,$ these order quantities are respectively denoted by EOQ$1$ and EOQ$2.$ The retailer finds out that $1000<$ EOQ$1<$ EOQ$2.$

Which one of the following is incorrect?

$($a$)$ c$1>$ c$2$

$($b$)$ The optimal order quantity for the retailer is EOQ$2$

$($c$)$ The optimal order quantity for the retailer is greater than $1000$

$($d$)$ None of the above