Consider the IEEE double precision format, which has $11$ bits allocated for

storing the exponent and $52$ bits allocated for storing the mantissa.

$($a$)$ Find the largest integer k for which fl$(19+2^-$k$)>$ fl$(19)$ in this system.

$($b$)$ In this system, what is the result of the computation $[1+(2^-51+2^-53)]-1?$

$($c$)$ In this system, what is the result of the computation $[1+(2^-51++2^-52+2^-53)]-1$