What happens when the same action extremal value can be obtained in more than one path in the configuration space?

I’m trying to understand the logic underneath the concept of action and lagrangian. I know this kind of questions have been asked many times, but I was unable to find an answer to this one.

I’ve ever read that the action should be minimized, or at least put in an extremum, in the actual physical trajectory of the system in the configuration space. I’ve also read that this has to do with a sort of "phase interference" between trajectory having the same non-extremal value of the action.

But what make us sure that an extremal value will exists? And supposing it exists why should be it unique?

I can for example imagine a Lagrangian dependent on two generalized coordinates and not explicit dependent on the parameter $s$, so that $Lequiv Lleft(q_1(s),q_2(s)right)$, but I can choose an initial and a final state $left(q_1(s_i),q_2(s_i)right),left(q_1(s_f),q_2(s_f)right)$, so that apparently there is not a unique path that minimizes the value of the integral
begin{equation*}
int_{s_i}^{s_f}
text{d}s Lleft(q_1(s),q_2(s)right)
end{equation*}
such as in the following picture,in which "everything is symmetric" between the two points $left(q_1(s_i),q_2(s_i),Lleft(q_1(s_i),q_2(s_i)right)right),left(q_1(s_f),q_2(s_f),Lleft(q_1(s_f),q_2(s_f)right)right)$, so that I can imagine two paths (not represented) with the same minimum value of the action.