A rectangular furnace has $2$ end black surfaces $($top and bottom$)$ and surface $1$ is kept at $400$ K and surface $2$ is kept at $800$ K $..$ The dimensions of the furnace are $22l($height$),$ Its lateral surfaces, the $4$ vertical walls, $($surface $3),$ are also black and kept at $1000$ K $.$ Answer the following questions:

$($a$)$ What is the view factor $F_{12}?$

$($b$)$ Draw the radiation network for this problem

$($e$)$ What is the heat transfer from the furnace $($surf $3)$ to surface $1?$

$($d$)$ What is the beat mate that need to be supplied to the lateral surface $($surface$3)$ to maintain this system?