On January $1,$ year $1,$ a company purchased machinery at a cost of $$200,000.$ Book depreciation was calculated using the straight$-$line balance method over an estimated useful life of $20$ years and no salvage value. The company recognized a full year of depreciation in the year of acquisition. The fair market value is the same as the book value if not stated otherwise. The company$’$s policy is to maximize the book value of the machinery.

Based on a year $2$ engineering study, a $$30,000$ impairment loss was recognized at the end of year $2($after depreciation is recognized$).$ Based on a year $4$ engineering study, the fair market value of the machinery was determined as $$155,000$ at the end of year $4($after depreciation is recognized$).$

Determine the book value of the machinery on December $31$ in