A topological space $X$ is called

$\bullet$ sequential if for each non-closed subset $A\subset X$ there exists a sequence $\{a_n\}_{n\in\omega}\subset A$ that converges to a point $a\notin A$;

$\bullet$ almost sequential if each points $x\in X$ is contained in a dense sequential subspace of $X$.

Question. Is there an almost sequential regular space $X$ which is not sequential (and moreover, contains a closed countable subspace $F\subset X$ that has no non-trivial convergent sequences)?

A topological space $X$ is called

$\bullet$ sequential if for each non-closed subset $A\subset X$ there exists a sequence $\{a_n\}_{n\in\omega}\subset A$ that converges to a point $a\notin A$;

$\bullet$ almost sequential if each point $x\in X$ is contained in a dense sequential subspace of $X$.

Question. Is there an almost sequential regular space $X$ which is not sequential (and moreover, contains a closed countable subspace $F\subset X$ that has no non-trivial convergent sequences)?