Millie is an employee of a large software company but also does consulting as an independent contractor. Last year, Millie's total tax liability was $$30,000.$ This year, she estimates her total tax liability will be $$42,000.$ Millie's AGI is $$145,000.$ Her employer will withhold $$22,000$ of tax from her paychecks throughout the year. What is the minimum amount Millie needs to pay through estimated tax payments to avoid underpayment penalties?

a$)$ $$0,$ she does not need to pay estimated tax payments if she has taxes withheld from her paychecks

b$)$ $$15,800,$ assuming her estimate of the current year tax liability is accurate

c$)$ $$8,000$

d$)$ $$11,000$