Digital Tech Dynamics purchased a new quality inspection system for $$550,000.$ The estimated salvage value was $$50,000$ after $10$ years. Currently, the expected remaining life is $7$ years with an AOC of $$50,000$ per year and an estimated salvage value of $$40,000.$ The new president has recommended early replacement of the system with one that costs $$360,000$ and has a $12-$year economic service life, a $$35,000$ salvage value, and an estimated AOC of $$50,000$ per year. If the MARR for the corporation is $12\%$ per year, use factor$-$based relations to determine the minimum trade$-$in value necessary now to make the president's replacement economically advantageous.

The minimum trade$-$in value necessary now to make the president's replacement economically advantageous is $