Solving this question will makes us realize that y-component of tension $T$ of the string satisfies the inequality
$0<T_y<0.671W$
And the y component of the force exerted by pivot $B$ on the rod $BC$ satisfies the inequality
$0<B_y<0.5W$
Which tells us
$$R_a>2W-0.671W-0.5W=0.829W$$
$$R_a<2W-0W=2W$$
$$0.829W<R_a<2W$$
Where $R_a$ is the force exerted by the pivot on the rod $AB$.
My question is how can the pivot exert an upward force on the rod AB? Is the rod glued with the pivot? How can a pivot provide a force in almost any direction to maintain equilibrium?
The definition of a pivot is that it is fixed in location and will provide whatever force is necessary to stay there. If the thing attached to it wants to move down it will provide the upward force to counter that. In this case, the two rods weigh $W$ each and will fall under gravity if unsupported. The pivot has to provide the support, so has to provide an upward force of $2W$. That way the net force on the triangle is zero and the system can be stable.