$44.$ When a gas expands along path AC shown below, it does $400$ J of work and absorbs either $200$ or $400$ J of heat. $($a$)$ Suppose you are told that along path ABC, the gas absorbs either $200$ or $400$ J of heat. Which of these values is correct? $($b$)$ Give the correct answer from part $($a$),$ how much work is done by the gas along $\backslash ($ A B C $\backslash )?($c$)$ Along CD$,$ the internal energy of the gas decreases by $50$ J $.$ How much heat is exchanged by the gas along this path?