When the system is taken from state $\text{'}$a$\text{'}$ to state $\text{'}$b$\text{'}$ along the path acb, $800J$ of heat flows into the system and the system does $300J$ of work.

$($a$)$ Suppose that the system is taken from $\text{'}$a$\text{'}$ to $\text{'}$b$\text{'}$ along the path adb, how much heat flows into the system given that the work done by the system is $120J?$

$($b$)$ Suppose that the system is returned from $\text{'}b\text{'}$ to $\text{'}$a$\text{'}$ directly by path $1.$ The work done on the system is $220J.$ Does the system absorb or liberate heat during the return? How much?