There exist a constant $c$ and an algorithm A such that for every Yes$-$instance $x$ of the problem $P,$

there is a string $s$ of length at most $|x{|}^c$ with the property that A outputs Yes on input $(x,s).$

Which of the following statements are negations of the above statement.

There exist a constant $c$ and an algorithm A such that for every Yes$-$instance $x$ of the

problem $P,$ there is a string $s$ of length at most $|x{|}^c$ with the property that A does not

output Yes on input $(x,s).$

There exist a constant $c$ and an algorithm A such that for every Yes$-$instance $x$ of the

problem $P,$ and every string $s$ of length at most $|x{|}^c,$A does not output Yes on input

$(x,s).$

For every constant $c$ and algorithm $A,$ there exists a Yes$-$instance $x$ of the problem $P,$ such

that for any string $s$ of length at most $|x{|}^c,$A does not output Yes on input $(x,s).$

For every constant $c,$ algorithm $A,$ Yes$-$instance $x$ of the problem $P,$ and string $s$ of length

at most $|x{|}^c,$A does not output Yes on input $(x,s).$

For every constant $c,$ algorithm $A,$ Yes$-$instance $x$ of the problem $P,$ and there exists a

string $s$ of length at most $|x{|}^c$ such that A does not output Yes on input $(x,s).$

For every constant $c,$ there exists an algorithm A such that for all Yes$-$instance $x$ of the

problem $P,$ and string $s$ of length at most $|x{|}^c,$A does not output Yes on input $(x,s).$