$,$ Consider the following network $G=(V,E)$ with source $s$ and sink $t$ :

$($a$)$ Run the Ford$-$Fulkson algorithm on this instance. For each iteration, illustrate the flow $f($i$.$e$.,f(e)$ for every einE, the augmenting path $p,$ the bottleneck capacity $x(p),$ and the value of the flow at the end of the iteration.

$($b$)$ Let $f^{**}$ to be the maximum$-$flow found in $($a$).$ Give the $s-t$ cut $(S^{**},T^{**}),$ where $s$ can reach $v$ in $\{$:$G_{f^{**}}\},$ and $T^{**}$:$=\frac{V}{\frac{?}{?}}{S}^{**}.$ Show the cut$-$edges and the capacity of this $s-t$ cut.

$($c$)$ Let $f^{**}$ to be the maximum$-$flow found in $($a$).$ Give the $s-t$ cut $(S^{\text{'}},T^{\text{'}}),$ where $v$ can reach $t$ in $\{$:$G_{f^{**}}\},$ and $S^{\text{'}}$:$=\frac{V}{\frac{?}{?}}{T}^{\text{'}}.$ Show the cut$-$edges and the capacity of this $s-t$ cut.