Sheridan Company is concerned about the accuracy of its year$-$end inventory balance. Inventory shows a year$-$end balance of $.$ Discussions with the company accountant reveal the following.

$1.$ Sheridan received goods costing on January $2$ that were shipped FOB destination on December $29.$ The shipment was a rush order that was supposed to arrive on December $31.$ This purchase was included in the ending inventory of $.$

$2.$ Sheridan sold goods costing to Cusa Company, FOB shipping point, on December $28$ for $.$ The goods are not expected to arrive at Cusa until January $12.$ The goods were not included in the physical inventory because they were not in the warehouse.

$3.$ The physical count of the inventory did not include goods costing that were shipped FOB destination to Sheridan on December $27$ and were still in transit at year$-$end.

$4.$ Sheridan received goods costing on January $2.$ The goods were shipped FOB shipping point on December $26$ by Noble Co$.$ The goods were not included in the physical count.

$5.$ Sheridan sold goods costing to Limerick Co$.$ for $.$ The goods were shipped FOB destination on December $30.$ The goods were received by Limerick on January $8$ and were not included in Sheridan's physical inventory.

$($a$)$ Determine Sheridan's correct inventory amount on December $31.$ Correct inventory amount

$($b$)$ Using the perpetual inventory system what correcting entry would have be made for item $4?$