Step by step $3$ Fixed Points

Consider the problem of determining if a program $P$ has any fixed points. Given any program $P,$ a fixed point is an input $x$ such that $P(x)$ outputs $x.$

$($a$)$ Prove that the problem of determining whether a program has a fixed point is uncomputable.

$($b$)$ Consider the problem of outputting a fixed point of a program if it has one, and outputting "Null" otherwise. Prove that this problem is uncomputable.

$($c$)$ Consider the problem of outputting a fixed point of a program $F$ if the fixed point exists and is a natural number, and outputting "Null" otherwise. If an input is a natural number, then it has no leading zero before its most significant bit.

Show that if this problem can be solved, then the problem in part $($b$)$ can be solved. What does this say about the computability of this problem? $($You may assume that the set of all possible inputs to a program is countable, as is the case on your computer.$)$