A healthcare provider offers a single service to its patients, and the patients are covered by only two different third parties. Payer $1,$ Medicare, represents $50\%$ of the patients and pays a fixed fee of $$415$ per patient. Payer $2,$ a Commercial Insurer, represents $45\%$ of the patients and pays $80\%$ of the provider's gross charge. The remaining patients are Charity Care patients who do not pay for their services. The provider treats $3,400$ patients per month. The fixed costs of the provider each month is $$79,000,$ and their variable cost per patient is $$425.$ The provider desires to set its gross charge per patient to achieve a profit, or net income, of $$92,000$ for the month.

a$.$ Set up an Algebraic expression, with $"p"$ being the gross charge $($price$)$ of the service to each patient being seen in the month, to solve this problem. Show this expression on your spreadsheet.

b$.$ Solve the problem finding the Gross Price to be charged to achieve the profit target.

c$.$ How many charity patients does the organization treat per month?

d$.$ What is the grand total sum of payments that the Commercial Insurer pays for all its patients in a month? $($Note$,$ this is not the total for one patient, but instead it is the total for all their patients together in the month$).$