$52.$ Kate acquires a principal residence in $2018$ for $$200,000$ secured by a $$180,000$ mortgage. In $2023,$ Kate misses work for three months. Her employer stops paying her after ten days. In $2023$ when the mortgage balance is $$170,000,$ Kate and the bank revise her terms and lower the loan balance to $$150,000.$ The market value of the home remains above $$200,000.$ How much cancellation of debt income will Kate recognize?

a$.$ $$0$

b$.$ $$20,000$

c$.$ $$10,000$

d$.$ $$150,000$

e$.$ $$170,000$