$($b$)$ Assume that we are given an implementation $A$ of Dijkstra's algorithm with worst$-$case time complexity $T(m,n),$ where $m$ and $n$ denote the number of edges and vertices in the input graph, respectively. You may assume that if $m^{\text{'}}m$ and $n^{\text{'}}n,$ then $T(m^{\text{'}},n^{\text{'}})T(m,n).$ You may also assume that $T(m,n)=(m+n).$ Explain how to use $A$ as a black box subroutine to correctly solve any SSSP instance with exactly one negative$-$weight edge and no negative$-$weight cycles in $O(T(m,n))$ time.

$($c$)$ Repeat part $($b$),$ but for SSSP instances with exactly $100$ negative$-$weight edges and no negative$-$weight cycles. As in part $($b$),$ your algorithm is required to run in $O(T(m,n))$ time. Hint: Note that expressions like $100,2^{100},$ and $100!$ are all $O(1).$