$[2]$ Consider the Ericsson Cycle learned in class. Air standard assistance and cold air standard assistance are applied. Each course consists of the following and is all reversible.

$1-2$: When the temperature is constant, the back is in Pyeongchang $($energy inflow$)$

$2-3$: Energy release under constant pressure

$3-4$: Compression of back $($energy release$)$ under constant temperature

$4-1$: Energy inflow with constant pressure

The working fluid is air and the working flow rate is $1$ kg$.$ It operates at $27$C and $527$C and has a maximum pressure of $2$ MPa and a pressure at $3$ points is atmospheric pressure $(100$ kPa$).$ The gas constant of air is $0\mathrm{.}287$ kJ$/$kgK$.$ The static specific heat and static pressure specific heat are $0\mathrm{.}7$ kJ$/$kgK and $0\mathrm{.}987$ kJ$/$kgK$,$ respectively.

Make sure to include the unit when you write the answer, indicate the energy supply including $,$ and indicate the efficiency in percentage $(24$ points$)$

$(1)$ Use the ideal gas equation to find the volume at $1$ point. $($To solve the problem below, find the volume at each point.$)$

$(2)1-2$: Apply the law of conservation of energy to find the amount of heat supplied.

$(3)2-3$: Apply the law of conservation of energy to find the amount of heat released.

$(4)3-4$: Apply the law of conservation of energy to find the amount of heat released.

$(5)4-1$: Apply the law of conservation of energy to find the amount of heat supplied.

$(6)$ Get pure work.

$(7)$ Find the efficiency in the absence of regeneration.

$(8)$ If the amount of heat released in the process of $2-3$ is supplied by $4-1($Renewable Ericsson Cycle$),$ find the efficiency.