Springs and Pulleys - Force analysis

[Null_Void]

Homework Statement

In the setup shown, a block is placed on a frictionless floor, the cords and pulleys are ideal and each spring has stiffness k. The block is pulled away from the wall. How far will the block shift, while the pulling force is gradually increased from zero to a value F

Pic below.

Relevant Equations

$F_s = Kx$

So I labelled all the springs from 1 to 4. Since the force is gradual, at equilibrium :

$F = k(x_2+ x_4) - - - - (1)$

Now we need to relate the different extensions of the springs:

$kx_1 = kx_4$ (same string)
$x_1 = x_4 (2)$
$kx_2 = 2kx_1$ ( ideal pulley)
$x_2 = 2x_1 (3)$

$kx_3 = 2kx_4$ (ideal pulley)
$x_3 = 2x_4$

Now I came to these equations by displacing each spring and assuming how much the pulleys would move. I'm not sure of how the string length would be distributed over the system.
I assumed that as spring-1 moves $x_1$ Half of this would be carried over to spring-2 and so on. I would like to know whether this is the right approach, and if so why?

Then I proceeded to equate $(1)$ with $(2)$ and $(3)$:

$F = k(3x_4)$
Thus,
$x_4 = F/3k$

Now this part is where I'm absolutely shaky:
Displacement of block $x = x_1/2 + x_2$

And so equating everything we get:

$x = 5{x_4}/2$

Therefore,

$x = 5F/6k$

But the given answer is ##10F/9k#

So where am I going wrong? And is there any way to easily deduce what would happen in such systems?

Thanks in Advance!

 

[haruspex]

Consider the three horizontal sections of string.
What is the relationship between the changes in lengths of the top and middle section and $x_1$ and $x_3$?

 

[Null_Void]

Consider the three horizontal sections of string.
What is the relationship between the changes in lengths of the top and middle section and $x_1$ and $x_3$?

The change in length of the top string is equal to the elongation of spring-1 while for the middle section it is equal to twice the elongation of spring-3