On April $30,2024,$ Run Company paid $\backslash (\backslash$$ $8,200,000\backslash )$ to acquire all of the common stock of Jump Corporation, which became a division of Run. Jump reported the following balance sheet at the time of the acquisition:

It was determined at the date of the purchase that the fair value of the identifiable net assets of Jump was $\backslash (\backslash$$ $7,800,000\backslash ).$ At December $31,2024,$ Run reports the following balance sheet information:

It is determined that the fair value of the Jump division is $\backslash (\backslash$$ $4,200,000\backslash ).$

Required

$1.$ Compute the amount of goodwill recognized, if any, on April $30,2024.$

$2.$ Determine the impairment loss, if any, to be recorded on December $31,2024.$ If an impairment loss is not to be recorded, explain why.

$3.$ Assume that the fair value of the Jump division is $\backslash (\backslash$$ $3,700,000\backslash )$ instead of $\backslash (4,200,000\backslash ).$ Prepare the journal entry to record the impairment loss, if any, on December $31,2024.$ If an impairment loss is not to be recorded, explain why.