The other answers suggest the use of FEA software. However this answer will instead focus on an analytical approach.
One of the main difficulties of the problem you are try to solve is that it is a statically indeterminate problem. In other words, the structure is more constrained than simpler structures (such as a simple cantilever) such that equilibrium of forces and moments is not enough to solve the structure. This is because, when using equilibrium only, there are more unknown variables than equations. More information is required.
So, to solve a statically indeterminate problem, compatibility must also be enforced, i.e. the structure needs to make sense geometrically. This will make sense later. To enforce compatibility, a method called the Force Method will be employed.
First of all, we need to convert the statically indeterminate problem into a statically determinate problem by making cuts at arbitrary and appropriate locations, simplifying the geometry. So this:
becomes this:
However, by making these cuts, it makes it seem as if the structure does not interact with each other at the sites of the cuts, like the way cutting a rope means one cut end no longer interacts with the other cut end, allowing the rope to be pulled apart. However, this is not the case, and so we need to add forces and bending moments at the sites of the cuts we have made to account for the interaction of the structure at these regions. In the example of a rope, this is akin to adding tension forces at the cut ends. These 'cut' forces are internal forces, and so, at every cut, the forces and bending moments must be equal and opposite. So the forces and moments to add back are:
At this moment in time, the values of these forces to add back are unknown variables. So, let us add up all the forces up (the externally applied force in our newly formed statically determinate structure + the cut forces) to obtain three cantilevers:
So, you can now easily determine the deflections of these three cantilevers individually, like you would for any other simple cantilever, in terms of $F$, plus the unknowns, $P_1, P_2, M_1, M_2$. The problem now is, how do you solve for the four unknowns? We need four equations after we have exhausted our use of equilibrium! This is the part where you actually enforce compatibility, by using 'compatibility conditions'. In this case, these are the equation forms of saying "we need each of the cantilevers to bend such that the ends are still the right distance apart from each other". Because you have a rigid plate connecting the three ends, you know the distance between the ends will not change. So we obtain two 'compatibility conditions':
$$\delta_{1} = \delta_{2}$$
$$\delta_{2} = \delta_{3}$$
Where $\delta$ is the deflection of each of the cantilevers (1,2,3) at the point the cantilever is connected at the plate. Also, note that the cantilevers must have the same rotation at the point connecting the plate, otherwise the rigid plate cannot be straight:
This provides the remaining two required compatibility conditions:
$$\theta_{1} = \theta_{2}$$
$$\theta_{2} = \theta_{3}$$
Where $\theta$ is the rotation of each cantilever at the point connecting it to the plate.
Now you easily determine an expression for the deflections of the system as desired!
