Consider the borrowing costs in USD faced by the following three companies:

Fixed Floating

A $4\mathrm{.}5\%$ SOFR $+0\mathrm{.}6\%$

B $6\mathrm{.}0\%$ SOFR $+1\mathrm{.}7\%$

C $5\mathrm{.}1\%$ SOFR $+1\mathrm{.}0\%$

Assume that if any two companies enter the swap transaction, they split the possible savings equally.

a$)$ Company A and company B want to engage in the swap transaction. Find the range for the swap rate within which both companies would benefit from the swap.

Suppose company C wants to borrow fixed$-$rate funds. Is it possible for C to reduce its cost of borrowing below $5\mathrm{.}1\%,$ and if so$,$ what is the lowest possible cost it could achieve?

c$)$ Suppose company C wants to borrow floating$-$rate funds. Is it possible for C to reduce its cost of borrowing below SOFR $+1\%,$ and if so$,$ what is the lowest possible cost it could achieve?