$3)[10$ points$]$ Suppose we want to estimate the number of hosts in a subnet behind its one$-$hop router using NAT. The IP layer stamps an identification number sequentially on each IP packet

generated by a host in the subnet, where the identification number of the first IP packet generated by a host in the subnet is a randomly chosen value of $32$ bits, and the identification numbers of the subsequent IP packets generated by that host are sequentially assigned. For example, if the first packet by host A is assigned the identification number $\backslash ($ r $\backslash )$ where $\backslash ($ r $\backslash )$ is a random number of $32$ bits, then the second packet generated by the host will be assigned with an identification number $\backslash ($ r$+1\backslash ),$ and so on$.$ As we know that all IP packets generated by the hosts in the subnet behind the NAT are sent to the outside world through the one$-$hop router, assume that the packet generation probability of each host in the subnet is almost equal,

$($a$)$ develop a simple method to estimate the number of hosts in the subnet behind a NAT, and justify why the method works. $(5$ points$)$

$($b$)$ Given a monitoring period, assuming that the number of packets from the outside world destined to each host in the subnet is almost identical, does the proposed method in Part $($a$)$ still work? Justify your answer. $(5$ points$)$