Paige creates a length$-$checking program which takes as input any list of natural numbers and outputs

True if the list has less than $100$ items in it$,$ and False if it has $100$ or more items in it$.$

She finds that her program always takes:

$5$n$^4+8$n $+3$ seconds when it returns True,

$9$n$^5+3$n$^2+1$ seconds when it returns False,

where n is the length of the input list.

$($A$)$ Does Paige$$s program have time complexity O$($n$^4)?$ State $$Yes$$ or $$No$$ and justify your answer.

$($B$)$ Does Paige$$s program have time complexity O$($n$^5)?$ State $$Yes$$ or $$No$$ and justify your answer.

$($C$)$ Does Paige$$s program have time complexity O$($n$^6)?$ State $$Yes$$ or $$No$$ and justify your answer.