XYZ Corporation expects the following sales for the next three months: Month $1-$ $$500,000,$ Month $2-$ $$600,000,$ Month $3-$ $$700,000.$ The company's collection pattern is $60\%$ in the month of sale, $30\%$ in the following month, and $10\%$ two months after the sale. What are the expected cash collections for Month $2?$

a$.$ $$510,000$

b$.$ $$570,000$

c$.$ $$600,000$

d$.$ $$630,000$