DEF Inc. has beginning cash of $$100,000.$ It expects sales of $$300,000$ in Month $1,$ $$350,000$ in Month $2,$ and $$400,000$ in Month $3.$ The company's collection pattern is $40\%$ in the month of sale, $30\%$ in the following month, and $30\%$ two months after the sale. Purchases are expected to be $$150,000$ in Month $1,$ $$180,000$ in Month $2,$ and $$200,000$ in Month $3.$ The company expects to pay $50\%$ of its purchases in the month of acquisition and the remaining $50\%$ in the following month. What is the ending cash balance for Month $1?$

a$.$ $$180,000$

b$.$ $$145,000$

c$.$ $$200,000$

d$.$ $$210,000$