Suppose we have an algorithm for the $3-$Coloring problem of the following form,

where "SolveX" is any algorithm for solving some other problem X$.$

Suppose further that we have proved that this algorithm is correct $($assuming the

SolveX algorithm correctly solves problem $x).$

Given this information $($in addition to what we already know about $3$Coloring in

general$),$ which of the following statements are correct?

$($The exact right option$($s$)$ must be chosen for this problem to count as solved; no

partial points.$)$

$3-$Coloring must be in $P.$

$3-$Coloring must be NP$-$hard.

Problem $x$ cannot possibly be in $P$

Problem X must be NP$-$hard.

None of the above.

Can you explain why we chose these alternative, and what I need to think of when I encounter like this problem in order to solve it$.$